mm-1099
===
Subject: Re: The joy of plain text
> My impression is that the most generally preferred method
of writing
> mathematics on this newsgroup is to use a simpli\fied version
of TeX,
> removing all symbols (such as dollars) that are unnecessary
for
> comprehension - for example:
> alpha^2 + beta^{5/2} = sum_{i=0} ^infty gamma_i ^{-3}.
> is not dif\ficult to read. How would you prefer that to be
written?
My sentiment more or less exactly! Everybody understands TeX
(or can
guess the meaning). My practice is to also leave out the
backslashes
(ÔÕ), if it looks like that wonÕt \
lead to any
misunderstandings, so I
might write:
alpha^2 + beta^{5/2} = sum_{i=0}^infty gamma_i^{-3}.
I think that this is a reasonable compromise and IMVHO
slightly
more readable. I also prefer not to use Ôfrac\
or Ôover(me
the
plainTeX-fan:) at all. I think that {daadaa}/{doobedoo} is
better
than the alternative ways of writing a quotient:)
I do feel that cutting and pasting from TeX-source has
certain other
drawbacks. E.g. if I were to cut and paste the above from a
TeX-\file
I had written, it probably wouldnÕt have that extra space
surrounding
Ôplusand Ôequal \
tosigns. I feel that this extra space
does enhance
legibility a bit, so I would normally do it that way. Ok.
Sometimes
I relax on that rule, if IÕm in a hurry and \
donÕt have the
time to edit
or proofread my postings.
But to summarize: Degustibus non est disputandum.
Jyrki Lahtonen, Turku, Finland
===
Subject: Re: The joy of plain text
<comqqh$smc$1@wisteria.csv.warwick.ac.uk>
<comugh$dd0$1@bowmore.utu.\fi> alpha^2 + beta^{5/2} = sum_{i=0}
^infty gamma_i ^{-3}.
> is not dif\ficult to read. How would you prefer that to be
written?
> I might write:
> alpha^2 + beta^{5/2} = sum_{i=0}^infty gamma_i^{-3}.
Much easier to read.
Other hard to read stuff is
ax^2+bx+c=(x-r)(x-s)=x^2-(r+s)x+rs=hardtoread
===
Subject: Re: analysis question
>Let 0 be a point of Lebesgue density of E subset mathbb{R}
(i.e.
>lim_{m(B) -> 0} m(B cap E)/m(B) = 1, where the limit is
taken over all
>balls about 0).
>Prove that there exists an in\finite sequence of points x_n in
E, with x_n
>!= 0, and also x_n -> 0 as n -> in\finity,
>such that the sequence also satis\fies -x_n in E and 2x_n in
E, for all
n.
Hint: 0 will also be a point of density of -E and of 1/2 E.
Make sure
that m(B cap E)/m(B) and m(B cap (-E))/m(B) and m(B cap (1/2
E)/m(B)
are large enough, and youÕll be able to ...
Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
===
Subject: Re: analysis question
>Let 0 be a point of Lebesgue density of E subset mathbb{R}
(i.e.
>lim_{m(B) -> 0} m(B cap E)/m(B) = 1, where the limit is
taken over
all
>balls about 0).
>Prove that there exists an in\finite sequence of points x_n in
E, with
x_n
>!= 0, and also x_n -> 0 as n -> in\finity,
>such that the sequence also satis\fies -x_n in E and 2x_n in
E, for
all n.
> Hint: 0 will also be a point of density of -E and of 1/2 E.
Make sure
> that m(B cap E)/m(B) and m(B cap (-E))/m(B) and m(B cap (1/2
E)/m(B)
> are large enough, and youÕll be able to ...
IÕd only add that if x is a point of Lebesgue density of \
both
E and F, then
x is a point of Lebesgue density of E * F, where * denotes
intersection. So
0 is a point of Lebesgue density of E * (-E) * (E/2).
===
Subject: Convergence Question (just for fun)
I was doing some work the other day, and had to keep taking
the ceiling of
half of a value. Eventually, I found that I was taking the
ceiling of half
of the ceiling of half of a value. And so on.
This led me to think about the following sequence of
functions, and whether
it converges or not.
f_1(x) = ceil(x/2)
f_2(x) = ceil(ceil(x/2)/2) = ceil(f_1(x)/2)
f_3(x) = ceil(ceil(ceil(x/2)/2)/2) = ceil(ceil(f_1(x)/2)) =
ceil(f_2(x)/2)
...
f_n(x) = ceil(f_n-1(x)/2)
...
It is not quite the same as
g_n(x) = x/(2^n)
I was just wondering if it converged.
- Tim
--
Timothy M. Brauch
NSF Fellow
Department of Mathematics
University of Louisville
email is:
news (dot) post (at) tbrauch (dot) com
===
Subject: Re: Convergence Question (just for fun)
> I was doing some work the other day, and had to keep taking
the ceiling
> of half of a value. Eventually, I found that I was taking
the ceiling of
> half of the ceiling of half of a value. And so on.
> This led me to think about the following sequence of
functions, and
> whether it converges or not.
> f_1(x) = ceil(x/2)
> f_2(x) = ceil(ceil(x/2)/2) = ceil(f_1(x)/2)
> f_3(x) = ceil(ceil(ceil(x/2)/2)/2) = ceil(ceil(f_1(x)/2)) =
> ceil(f_2(x)/2) ...
> f_n(x) = ceil(f_n-1(x)/2)
> ...
> It is not quite the same as
> g_n(x) = x/(2^n)
> I was just wondering if it converged.
It converges to
0 if x <= 0,
1 if x > 0
(which is similar to the Heaviside unit step function).
David
===
Subject: Re: Convergence Question (just for fun)
> It converges to
> 0 if x <= 0,
> 1 if x > 0
> (which is similar to the Heaviside unit step function).
> David
Sorry, I realized that if it converged, it must converge to
above. I
guess, I was really wondering what kind of convergence does
it have?
Given any \fixed n, I can always \find an x such \
that f_n(x) > 1.
Thus it is
not uniform convergence. Other than that, I am stumped as to
what kind of
convergence it does exhibit.
- Tim
--
Timothy M. Brauch
NSF Fellow
Department of Mathematics
University of Louisville
email is:
news (dot) post (at) tbrauch (dot) com
===
Subject: Re: Convergence Question (just for fun)
> It converges to
> 0 if x <= 0,
> 1 if x > 0
> (which is similar to the Heaviside unit step function).
> Sorry, I realized that if it converged, it must converge to
above. I
> guess, I was really wondering what kind of convergence does
it have?
> Given any \fixed n, I can always \find an x such \
that f_n(x) >
1. Thus it
> is not uniform convergence. Other than that, I am stumped
as to what
> kind of convergence it does exhibit.
This doesnÕt answer your question, but FWIW:
YouÕd said before that f_n(x) is not quite the same as \
g_n(x)
=
x/(2^n).
But f_n(x) is precisely the same as ceiling(x/(2^n)).
David
===
Subject: Proposed de\finition for comparing the sizes of two
sets
Proposed De\finition: A set Y is said to be larger than a set X
iff there
exists no function mapping X onto all of Y. (i.e. there is no
surjection
from X to Y)
Is this a workable de\finition that covers all cases? If so, is
it widely
used?
Dan
===
Subject: Re: Proposed de\finition for comparing the sizes of
two sets
at 08:57 PM, dchris@netcom.ca (Dan Christensen) said:
>Proposed De\finition: A set Y is said to be larger than a set
X iff
>there exists no function mapping X onto all of Y. (i.e.
there is no
>surjection from X to Y)
>Is this a workable de\finition that covers all cases? If so,
is it
>widely used?
Are you assuming the Axiom of Choice, or some equivalent? If
not, you
have to deal with incomparable pairs of sets.
--
Shmuel (Seymour J.) Metz, SysProg and JOAT
<http://patriot.net/~shmuel>
Unsolicited bulk E-mail subject to legal action. I reserve the
right to publicly post or ridicule any abusive E-mail. Reply
to
domain Patriot dot net user shmuel+news to contact me. Do not
===
Subject: Re: Proposed de\finition for comparing the sizes of
two sets
> at 08:57 PM, dchris@netcom.ca (Dan Christensen) said:
>Proposed De\finition: A set Y is said to be larger than a set
X iff
>there exists no function mapping X onto all of Y. (i.e.
there is no
>surjection from X to Y)
>Is this a workable de\finition that covers all cases? If so,
is it
>widely used?
> Are you assuming the Axiom of Choice, or some equivalent?
If not, you
> have to deal with incomparable pairs of sets.
I donÕt currently have AC built into my program, but you can
introduce
it as a premise at the beginning of any proof. I am planning
to make
it a true axiom, as well as building in some de\fintions for
cardinal
numbers in a future release. Some details need to be worked
out.
Dan
Download DC Proof 1.0 at http://www.dcproof.com
===
Subject: Re: Proposed de\finition for comparing the sizes of
two sets
>Proposed De\finition: A set Y is said to be larger than a set
X iff there
>exists no function mapping X onto all of Y. (i.e. there is
no surjection
>from X to Y)
>Is this a workable de\finition that covers all cases? If so,
is it widely
>used?
This is not a workable de\finition. Without at least some
form of AC, one can have too sets, each of which comes
out as larger than the other.
Consider a non-wellorderable set X and a well-orderable
set Y which is not smaller than or equal to (in the usual
sense) P(X). If there is a non-wellorderable set, this
must exist.
Now a surjection of X onto Y is equivalent to an injection
of Y into P(X), which we have assumed does not hold. And
a surjection of a well-orderable set onto any set gives a
well-ordering of that set.
--
This address is for information only. I do not claim that
these views
are those of the Statistics Department or of Purdue
University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
===
Subject: Re: Proposed de\finition for comparing the sizes of
two sets
> Proposed De\finition: A set Y is said to be larger than a set
X iff there
> exists no function mapping X onto all of Y. (i.e. there is
no surjection
> from X to Y)
Whoops forgot to include WilliamÕs Metatheorem in my other
post. A malady
which I now cure. There is no know proof of WMT nor has
anyone, not even
the mighty James Harris, been as yet able to produce a
counter example.
> Is this a workable de\finition that covers all cases? If so,
is it widely
> used?
Yes and since itÕs already well know, it also fails to be a
counter
example to:
Whatever math I dream up is already old hat.
-- WilliamÕs Metatheorem.
===
Subject: Re: Proposed de\finition for comparing the sizes of
two sets
> Proposed De\finition: A set Y is said to be larger than a set
X iff
there
> exists no function mapping X onto all of Y. (i.e. there is
no
surjection
> from X to Y)
> Whoops forgot to include WilliamÕs Metatheorem in my other
post. A malady
> which I now cure. There is no know proof of WMT nor has
anyone, not even
> the mighty James Harris, been as yet able to produce a
counter example.
> Is this a workable de\finition that covers all cases? If so,
is it
widely
> used?
> Yes and since itÕs already well know, it also fails to be \
a
counter
> example to:
> Whatever math I dream up is already old hat.
> -- WilliamÕs Metatheorem.
I have a version of CantorÕs Theorem included with my DC
Proof software
that
satis\fied with my explanation, but I began to wonder if
perhaps my
de\finition might need some work. So, I am actually relieved to
hear that is
Dan
Download DC Proof 1.0 at http://www.dcproof.com
===
Subject: Re: Proposed de\finition for comparing the sizes of
two sets
> Proposed De\finition: A set Y is said to be larger than a set
X iff
there
> exists no function mapping X onto all of Y. (i.e. there is
no
surjection
> from X to Y)
> Whoops forgot to include WilliamÕs Metatheorem in my other
post. A malady
> which I now cure. There is no know proof of WMT nor has
anyone, not even
> the mighty James Harris, been as yet able to produce a
counter example.
> Is this a workable de\finition that covers all cases? If so,
is it
widely
> used?
> Yes and since itÕs already well know, it also fails to be \
a
counter
> example to:
> Whatever math I dream up is already old hat.
> -- WilliamÕs Metatheorem.
I have a version of CantorÕs Theorem included with my DC \
Proof
questioning the proof. He was satis\fied with my explanation,
but I
began to wonder if perhaps my de\finition might need some work.
It is
not exactly the same as I have found elsewhere. So, I am
actually
Dan
Download DC Proof 1.0 at http://www.dcproof.com
===
Subject: Re: Proposed de\finition for comparing the sizes of
two sets
>[...]
> Whatever math I dream up is already old hat.
> -- WilliamÕs Metatheorem.
Actually that includes WilliamÕs Metatheorem. Sorry...
************************
David C. Ullrich
===
Subject: Re: Proposed de\finition for comparing the sizes of
two sets
> Proposed De\finition: A set Y is said to be larger than a set
X iff there
> exists no function mapping X onto all of Y. (i.e. there is
no surjection
> from X to Y)
> Is this a workable de\finition that covers all cases? If so,
is it widely
> used?
> Dan
It is essentially the de\finition Cantor used, and I believe it
covers
all cases provided one has the axiom of choice.
===
Subject: Re: Proposed de\finition for comparing the sizes of
two sets
> Proposed De\finition: A set Y is said to be larger than a set
X iff
there
> exists no function mapping X onto all of Y. (i.e. there is
no
surjection
> from X to Y)
> Is this a workable de\finition that covers all cases? If so,
is it
widely
> used?
> Dan
> It is essentially the de\finition Cantor used, and I believe
it covers
> all cases provided one has the axiom of choice.
There are de\finitions of relative set sizing besides
cardinality, for
example, there is a de\finition of set sizing that a proper
superset of a
set
is larger than the set, for \finite and in\finite \
sets, as
veri\fied by Katz.
As well, in the consideration of number theory, the set of
even integers,
as
an example, has half the asymptotic density of the integers,
within the
integers. Another notion of larger is X > Y when X has
in\finitely many
proper subsets that are proper supersets of Y, essentially a
stronger
condition than the proper superset size relation.
The asymptotic density is rather useful, for example, half of
the integers
are even integers, and twice as many integers are not
multiples of three as
are multiples of three.
For example, consider the power set of the naturals where
half of the
subsets
of the naturals have as an element a given element of the
naturals, and a
quarter have both of two elements. The set of all subsets of
the naturals
containing as an element a given element of the naturals is
half the
size
of the entire set.
One way to reinforce that intuitive concept is as so: compare
1/2 to 1/4.
Pick a subset of the set of all natural numbers at random,
the chance that
it
contains as an element zero is one half, and that it contains
18 zillion
even
is one half, and that it contains both is one fourth.
For disjoint sets, the above methods can work by comparing
each to their
union, or some other superset of their union, and then
comparing those
relative sizes.
There are probably yet other ways to meaningfully compare two
setssizes.
Using cardinality will probably get you a passing grade.
There is
considerable argument about whether in\finite sets
automatically have
bijections.
Ross F.
===
Subject: Re: Proposed de\finition for comparing the sizes of
two sets
> There are de\finitions of relative set sizing besides
cardinality, for
> example, there is a de\finition of set sizing that a proper
superset
> of a set is larger than the set
Are you referring to the partial order you get by using the
subset
predicate?
I.e. the question is whether S1 is a subset of S2?
Unfortunately that method leaves almost all pairs of sets
uncomparable.
For example consider two sets {a, b} and {a, c}. Neither is a
subset of
the other, so the two sets are uncomparable by the subset
method, even
though almost anyone can see that each has two elements so
they ought
to be considered the same size. The whole idea of cardinality
is to
clarify what the common de\finition means for \
\finite sets in a
way that
directly applies in the same way to non-\finite sets. Your
proposal of
using the subset property doesnÕt even give the right answer
for
\finite sets, so it fails at the desired task.
> in the consideration of number theory, the set of even
integers, as
> an example, has half the asymptotic density of the
integers, within
> the integers.
Do you know the difference between a set, which has no
properties other
than what are elements and what arenÕt elements, and a more
complicated
structure that has a set plus some additional operators
de\fined on the
elements of the set? The set of integers, and the set of even
integers,
have no such thing as asymptotic density when all you can
consider is
what elements are in the sets and what elements arenÕt.
Asymptotic
density can be de\fined only when you have some kind of metric
de\fined
between elements of the sets. If you use the real metric,
whereby the
distance between any two integers is the absolute value of
their
arithmetic difference, and as with any metric space you can
de\fine
neighborhoods of various radii, and you can de\fine various
kids of
limits with respect to that metric, such as the limit as the
size of
the ball from some \fixed point grows larger without limit,
then you can
de\fine asymptotic density in terms of those
metric-space-related terms.
But if you use a different metric on the very same set, such
as the
2-adic metric, you get a completely different result. So
obviously
asymptotic density isnÕt a property of the set itself, but
only a
property of one or another speci\fic metric space using the set
as
elements in the space.
> The asymptotic density is rather useful
Only in a metric space. In just set theory itÕs not even
de\fineable
(unless you use set theory to de\fine a metric space of course,
like the
way set theory is used to construct the natural numbers, from
which you
can construct the integers, from which you can construct the
rationals,
from which you can de\fine a metric, etc.).
===
Subject: Re: Proposed de\finition for comparing the sizes of
two sets
> There are de\finitions of relative set sizing besides
cardinality, for
> example, there is a de\finition of set sizing that a proper
superset
> of a set is larger than the set
> Are you referring to the partial order you get by using the
subset
predicate?
> I.e. the question is whether S1 is a subset of S2?
> Unfortunately that method leaves almost all pairs of sets
uncomparable.
> For example consider two sets {a, b} and {a, c}. Neither is
a subset of
> the other, so the two sets are uncomparable by the subset
method, even
> though almost anyone can see that each has two elements so
they ought
> to be considered the same size. The whole idea of
cardinality is to
> clarify what the common de\finition means for \
\finite sets in a
way that
> directly applies in the same way to non-\finite sets. Your
proposal of
> using the subset property doesnÕt even give the right
answer for
> \finite sets, so it fails at the desired task.
> in the consideration of number theory, the set of even
integers, as
> an example, has half the asymptotic density of the
integers, within
> the integers.
> Do you know the difference between a set, which has no
properties other
> than what are elements and what arenÕt elements, and a \
more
complicated
> structure that has a set plus some additional operators
de\fined on the
> elements of the set? The set of integers, and the set of
even integers,
> have no such thing as asymptotic density when all you can
consider is
> what elements are in the sets and what elements arenÕt.
Asymptotic
> density can be de\fined only when you have some kind of
metric de\fined
> between elements of the sets. If you use the real metric,
whereby the
> distance between any two integers is the absolute value of
their
> arithmetic difference, and as with any metric space you can
de\fine
> neighborhoods of various radii, and you can de\fine various
kids of
> limits with respect to that metric, such as the limit as
the size of
> the ball from some \fixed point grows larger without limit,
then you can
> de\fine asymptotic density in terms of those
metric-space-related terms.
> But if you use a different metric on the very same set,
such as the
> 2-adic metric, you get a completely different result. So
obviously
> asymptotic density isnÕt a property of the set itself, but
only a
> property of one or another speci\fic metric space using the
set as
> elements in the space.
> The asymptotic density is rather useful
> Only in a metric space. In just set theory itÕs not even
de\fineable
> (unless you use set theory to de\fine a metric space of
course, like the
> way set theory is used to construct the natural numbers,
from which you
> can construct the integers, from which you can construct
the rationals,
> from which you can de\fine a metric, etc.).
Do you have any in\finite sets? Please name an \
in\finite set
besides
the integers or reals.
The counting numbers, the natural integers, are very useful
for
counting these other sets of things. As each set contains
only unique
elements and there is an ordering relation on each of those
sets of
things, it is simple to see why each is numbered individually.
When the sets are disjoint, disjoint, or partially disjoint,
then
there exists a proper superset containing each element of
both.
YouÕll notice that the proper superset is larger than the \
set.
(YouÕll notice that was ignored.) You might see why you \
could
apply
that to any pair of sets, thus that it is universally
applicable.
About density and metrics, the metric is a great thing. IÕm
interested in this measure theory. For example, I consider
the sigma
algebra in relation to my rootsets and decorated ordinals, in
the
theory with ubiquitous ordinals where the ordinals are Z, the
integers, in the complete, concrete, and consistent theory. I
browse
the MathWorld de\finition of the sigma algebra and wonder what
he means
when Eric has sequences in the statement.
In terms of the sets where they are de\fined to be containing
the, for
example, integers, they are then those integers. The notion of
counting in any form, enumerating each, for example via a
choice
function or well ordering and induction, reßects back upon the
completely intuitive natural counting numbers.
ThatÕs especially so in the \finite case.
Besides the metrics there is also the intertwined notion of
probability distributions. Not about that, the set of even
integers
within the integers, is de\fined by the integral modulus.
Cardinality
has nothing to say about asymptotic density except no opinion.
Are half the in\finite binary sequences normal or through
restricted
sequence element interchange convertible to the canonical
sequence
with equal zero and one density, one third or two thirds?
The in\finitesimal predates the cardinal, in a sense being the
classical. The cardinal is in a sense a modern red herring.
The set
of all sets is its own powerset.
The in\finite is not necessarily simple nor intuitive except
that it
is: half of the integers are even. Bijections exist between
the
integers and even integers, and the particular one f(x)=2x, a
plain
straight line, shows that there are twice as many integers as
even
integers, in the integers or superset of the integers.
You have some good points there, but in comparing the
relative sizes
or in\finite sets there is sometimes a reason that has to do
with
solving a real world problem instead of ßights of fancy about
meaningless escapisms from foundational foundations.
Asymptotic
density is a useful notion that runs right back to one plus
one equals
two.
Ken, 2 + 2 = 4. To Scandinavians herring is a way of life, to
Americans itÕs a Monty Python sketch. I saw this the other
day, itÕs
haunting, yet funny: http://www.khaaan.com/ . It gets more
haunting
and less funny.
There is an implied point set topology and metric space, from
nothing,
wherefrom all is implied.
Ross Finlayson
===
Subject: Re: Proposed de\finition for comparing the sizes of
two sets
> Do you have any in\finite sets?
Please state what you mean when you use the word have. The
ordinary
meaning, synonym with own, isnÕt applicable, because sets \
are
abstract mathematical ideas which canÕt be owned by any one
person.
> Please name an in\finite set ...
Please state what you mean when you use the word name. The
ordinary
meaning, meaning to assign a name to an object, for example
naming a
baby, doesnÕt seem useful in this context. So \
thereÕs this
in\finite
set, and you want me to name it Fred??
> The counting numbers, the natural integers, are very useful
for
> counting these other sets of things.
Are you talking about using natural numbers (positive
integers) as
context-sensitive names for individual elements of sets, for
example in
the context of the set of rational numbers you can call 1/3
#1 and you
can call 4/7 #2 and you can call 0 #3 and you can call -5000
#4 etc.,
assigning a different natural-number label to each element in
the set?
Or are you talking about cardinality of \finite subsets of some
context-establishing set, for example the rationals, whereby
the subset
consisting of {1/3, 4/7} would be counted as size 2, and the
subset
consisting of {-5000, 0, 22/7, 1/2, 1/3} would be counted as
size 5?
> As each set contains only unique elements
I have no idea what you mean by a unique element. Every thing
is
unique, so of course every element of any set is a unique
thing hence a
unique element. Do you actually mean anything by what you
said?
> and there is an ordering relation on each of those sets of
things,
That is such a gross understatment that itÕs grossly
misleading hence
basically a lie. Not only is there one ordering relation on a
set, but
for any set containing at least two elements there are at
least two
different ordering relation on the set, and for any set
containing at
least three elements there are at least six different ordering
relations on the set, and for the integers there are an
uncountable
in\finity (aleph_null factorial, which equals aleph_one) of
different
ordering relations.
> completely intuitive natural counting numbers.
The natural numbers arenÕt completely intuitive. Only the
numbers from
one up to about four or \five are completely intuitive for
humans, where
they can just glance at a visual image showing that many
similar
objects in any random orientation and immediately know
intuitively,
without needing to count them, how many there are. Some birds
can
intuitively recognize cardinality up to about seven, beating
humans by
a couple, which is very useful for detecting if any eggs have
been
removed from the nest or added to the nest. If objects are in
standard
partterns, such as pips on a die-face or half-domino, then we
can
recognize them up to nine, but put those same pips in random
pattern
and suddenly the problem gets much more dif\ficult.
> the set of even integers within the integers, is de\fined by
the
> integral modulus.
Wrong! If you treat the integers as *nothing* except a set,
no order
relation, no arithmetic properties, and the individual
integers arenÕt
de\fined in terms of something else such as cardinality of sets
whereby
you can use that de\finition to generate arithmetic properties,
there is
no way whatsoever to de\fine which are even and which \
arenÕt
even except
by an in\finitely long list enumerating each and every even
integer (or
alternately by enumerating the complement set which are odd).
Here are four examples of sets of integers:
(1) Recursive de\finition: The empty set, and any set
containing exactly
one element which is an integer. Thus {} {{}} {{{}}} etc. are
the
consecutive integers. Even can be de\fined recursively like
this: N is
even iff N = {M} and M is not even. This works because
integers arenÕt
just abstract elements, but are actually constructed via set
theory in
a way such that even can be de\fined from that.
(2) Arithmetic de\finition: Start with PeanoÕs \
postulates, in
particular
the successor function S, with natural numbers de\fined
recursively as
1, and any S(N) where N is an integer. Even can be de\fined
recursively
like this: N is even iff N = S(M) where M is not even.
(3) Explicit listing of just a few integers because each one
is listed
separately and I donÕt have an in\finite amount \
of time to type
them
all: {apat, isa, lima, delawa, tatlo}. Unless you know that
IÕve used
Tagalog names for those \five integers, and unless you actually
know
what those \five Tagalog words mean, and unless \
IÕm actually
using the
corect Tagalog names instead of shufßing them, you canÕt
\figure out
which of those are odd and which are even in my set of
integers.
If I tell you that apat and delawa are even, and the other
three arenÕt
even, would you believe me? If I told you something else
instead, would
you believe me? On what basis could you decide for yourself
which are
even and which arenÕt per my de\finition unless I \
simply tell
you the
answer and promise not to change my de\finition to pull a trick
on you?
(4) In all the above, I had some sort of name for each
integer. But
suppose I donÕt have any names at all. Here are a bunch of
integers,
each displayed as an asterisk:
* * * * * * * * * *
* * * * * * * * * * * *
* * ** * * * * * * * *
Now suppose that I claimed that was a viewport into a small
section of
my integers, that really there are an in\finite number of them,
and they
are not displayed in any particular order that would make
sense to you,
but I promise I do have a grand design whereby that pattern
you see
above is one 3-by-62 portion of the overall design. Now how
would you
decide which of those asterisks are even numbers and which
arenÕt?
No matter how you guess which are even and which arenÕt, I
could
legitimately claim you are 100% wrong. If you get two
guesses, I could
legitimately claim you are 50% wrong, or worse, in each case.
Suppose I
told you an algorithm for deciding in each position whether
one of the
integers is there or that position is empty, not just for
that 3-by-62
viewport but for the entire grand pattern. How would you
de\fine even
vs. not-even for all the asterisks? Well because they are
located in a
lattice, you could perhaps make up a de\finition that is based
on the
location within the pattern. It wouldnÕt match my \
de\finition,
but at
least you *could* de\fine what is meant by even on that set of
integers. But that would not be de\fining even on the elements
themselves, rather youÕd be de\finining even \
based on their
position
within some sort of structue, in this case a regular lattice.
But
suppose they werenÕt in any lattice structure, but just
abstract
objects that you could get somehow, not in any particular
order, not in
any structure, no way to examine them internally, the only
predicates
you have are: (a) For any item, either that item is in the
set (of
integers) or not; (b) For any two items in the set, they are
either the
same element or not the same. With only those two predicates,
thereÕs
no way to de\fine even rigorously.
Suppose in some object-oriented language, for example Java, I
provided
for you a playpen whereby you could execute commands
interactively. I
provide for you the following functions/methods:
(1) static getInteger() ==> an integer object
(2) integerObject.toString() ==> SomeInteger (every integer
prints the
same)
(3) integerObject.hashCode() ==> 0 (every integer has the
same hashcode
too!)
(4) integerObject.equals(integer) ==> true if same object,
false otherwise
By calling getInteger() from time time, getting several such
integerObjects, can you decide just from calls to the above
API which
of them are even and which arenÕt? No, the best you can do \
is
make
arbitrary decisions, which wouldnÕt be consistent from one
run of the
demo to another.
Can you de\fine a predicate:
boolean isEven()
which is well de\fined, using *only* calls to the API \
IÕve
speci\fied above?
No, you canÕt. The best you can do is randomly decide for
each new
integerObject whether itÕs even or odd, and keep a cache of
such
decisions so you donÕt contradict yourself later, but \
thatÕs
not a
de\finition, thatÕs a random sampler, and again, \
just like the
manual
demo, youÕd get different results with each run of the
program, so your
function/method doesnÕt *de\fine* a function, it \
merely
generates a new
random sample each time itÕs run.
By the way, with *all* the integers, not just the positive
integers,
even if you know the total ordering, thatÕs still not enough
to de\fine
what is even and what isnÕt even. The best you can do is
divide the
ordered set into two subsets, alternating, and then pick one
at random
to be even. And itÕs even worse: If you have a playpen where
you can
call an API which gives you random integers and tells for any
two
integers whether they are equal and if not which is the
smaller of the
two, since you donÕt know whether two integers are adjacent
or not you
canÕt in a \finite number of steps determine the \
two
alternating sets.
> The set of all sets ...
ThereÕs no such thing. If you understood
proof-by-contradiction I could
present you a very simple proof to that fact, but you donÕt
seem to
understand even that simple aspect of mathematical logic so
it would be
a waste of my time to present it to you.
> half of the integers are even.
As just a set, 99% of them are even too.
> Bijections exist between the integers and even integers,
and the
> particular one f(x)=2x, a plain straight line, shows that
there are
> twice as many integers as even integers, in the integers or
superset of
> the integers.
It shows no such thing!! The bijection shows there are
exactly the same
number of integers as even integers. For every integer
thereÕs a
corresponding even integer, and vice versa.
What you said is equivalent to saying thereÕs a bijection
between the
\fingers on my left hand and the \fingers on my \
right hahd, which
shows
there are twice as many \fingers on my left hand as on my right
hand.
===
Subject: Re: Proposed de\finition for comparing the sizes of
two sets
> Do you have any in\finite sets?
> Please state what you mean when you use the word have. The
ordinary
> meaning, synonym with own, isnÕt applicable, because sets
are
> abstract mathematical ideas which canÕt be owned by any \
one
person.
> Please name an in\finite set ...
> Please state what you mean when you use the word name. The
ordinary
> meaning, meaning to assign a name to an object, for example
naming a
> baby, doesnÕt seem useful in this context. So \
thereÕs this
in\finite
> set, and you want me to name it Fred??
> The counting numbers, the natural integers, are very useful
for
> counting these other sets of things.
> Are you talking about using natural numbers (positive
integers) as
> context-sensitive names for individual elements of sets,
for example in
> the context of the set of rational numbers you can call 1/3
#1 and you
> can call 4/7 #2 and you can call 0 #3 and you can call
-5000 #4 etc.,
> assigning a different natural-number label to each element
in the set?
> Or are you talking about cardinality of \finite subsets of
some
> context-establishing set, for example the rationals,
whereby the subset
> consisting of {1/3, 4/7} would be counted as size 2, and
the subset
> consisting of {-5000, 0, 22/7, 1/2, 1/3} would be counted
as size 5?
Identify an in\finite set.
> As each set contains only unique elements
> I have no idea what you mean by a unique element. Every
thing is
> unique, so of course every element of any set is a unique
thing hence a
> unique element. Do you actually mean anything by what you
said?
Yes, I tend to be suf\ficiently exact.
> and there is an ordering relation on each of those sets of
things,
> That is such a gross understatment that itÕs grossly
misleading hence
> basically a lie. Not only is there one ordering relation on
a set, but
> for any set containing at least two elements there are at
least two
> different ordering relation on the set, and for any set
containing at
> least three elements there are at least six different
ordering
> relations on the set, and for the integers there are an
uncountable
> in\finity (aleph_null factorial, which equals aleph_one) of
different
> ordering relations.
ItÕs de\finitely not a lie. There are obviously \
many ordering
relations,
pick one.
> completely intuitive natural counting numbers.
> The natural numbers arenÕt completely intuitive. Only the
numbers from
> one up to about four or \five are completely intuitive for
humans, where
> they can just glance at a visual image showing that many
similar
> objects in any random orientation and immediately know
intuitively,
> without needing to count them, how many there are. Some
birds can
> intuitively recognize cardinality up to about seven,
beating humans by
> a couple, which is very useful for detecting if any eggs
have been
> removed from the nest or added to the nest. If objects are
in standard
> partterns, such as pips on a die-face or half-domino, then
we can
> recognize them up to nine, but put those same pips in
random pattern
> and suddenly the problem gets much more dif\ficult.
How many states are in the union? How many continents are on
the planet?
How many stars are in the sky?
> the set of even integers within the integers, is de\fined by
the
> integral modulus.
> Wrong! If you treat the integers as *nothing* except a set,
no order
> relation, no arithmetic properties, and the individual
integers arenÕt
> de\fined in terms of something else such as cardinality of
sets whereby
> you can use that de\finition to generate arithmetic
properties, there is
> no way whatsoever to de\fine which are even and which \
arenÕt
even except
> by an in\finitely long list enumerating each and every even
integer (or
> alternately by enumerating the complement set which are
odd).
> Here are four examples of sets of integers:
> (1) Recursive de\finition: The empty set, and any set
containing exactly
> one element which is an integer. Thus {} {{}} {{{}}} etc.
are the
> consecutive integers. Even can be de\fined recursively like
this: N is
> even iff N = {M} and M is not even. This works because
integers arenÕt
> just abstract elements, but are actually constructed via
set theory in
> a way such that even can be de\fined from that.
> (2) Arithmetic de\finition: Start with PeanoÕs \
postulates, in
particular
> the successor function S, with natural numbers de\fined
recursively as
> 1, and any S(N) where N is an integer. Even can be de\fined
recursively
> like this: N is even iff N = S(M) where M is not even.
OK. The powerset is the successor is the order type.
> (3) Explicit listing of just a few integers because each
one is listed
> separately and I donÕt have an in\finite amount \
of time to
type them
> all: {apat, isa, lima, delawa, tatlo}. Unless you know that
IÕve used
> Tagalog names for those \five integers, and unless you
actually know
> what those \five Tagalog words mean, and unless \
IÕm actually
using the
> corect Tagalog names instead of shufßing them, you \
canÕt
\figure out
> which of those are odd and which are even in my set of
integers.
> If I tell you that apat and delawa are even, and the other
three arenÕt
> even, would you believe me? If I told you something else
instead, would
> you believe me? On what basis could you decide for yourself
which are
> even and which arenÕt per my de\finition unless \
I simply tell
you the
> answer and promise not to change my de\finition to pull a
trick on you?
Do they mean the same thing as {1, 2, 3, 4, 5}?
> (4) In all the above, I had some sort of name for each
integer. But
> suppose I donÕt have any names at all. Here are a bunch of
integers,
> each displayed as an asterisk:
> * * * * * * * * * *
> * * * * * * * * * * * *
> * * ** * * * * * * * *
> Now suppose that I claimed that was a viewport into a small
section of
> my integers, that really there are an in\finite number of
them, and they
> are not displayed in any particular order that would make
sense to you,
> but I promise I do have a grand design whereby that pattern
you see
> above is one 3-by-62 portion of the overall design. Now how
would you
> decide which of those asterisks are even numbers and which
arenÕt?
I donÕt care. I wouldnÕt.
> No matter how you guess which are even and which arenÕt, I
could
> legitimately claim you are 100% wrong. If you get two
guesses, I could
> legitimately claim you are 50% wrong, or worse, in each
case. Suppose I
> told you an algorithm for deciding in each position whether
one of the
> integers is there or that position is empty, not just for
that 3-by-62
> viewport but for the entire grand pattern. How would you
de\fine even
> vs. not-even for all the asterisks? Well because they are
located in a
> lattice, you could perhaps make up a de\finition that is
based on the
> location within the pattern. It wouldnÕt match my
de\finition, but at
> least you *could* de\fine what is meant by even on that set \
of
> integers. But that would not be de\fining even on the \
elements
> themselves, rather youÕd be de\finining even \
based on their
position
> within some sort of structue, in this case a regular
lattice. But
> suppose they werenÕt in any lattice structure, but just
abstract
> objects that you could get somehow, not in any particular
order, not in
> any structure, no way to examine them internally, the only
predicates
> you have are: (a) For any item, either that item is in the
set (of
> integers) or not; (b) For any two items in the set, they
are either the
> same element or not the same. With only those two
predicates, thereÕs
> no way to de\fine even rigorously.
> Suppose in some object-oriented language, for example Java,
I provided
> for you a playpen whereby you could execute commands
interactively. I
> provide for you the following functions/methods:
> (1) static getInteger() ==> an integer object
> (2) integerObject.toString() ==> SomeInteger (every integer
prints
the same)
> (3) integerObject.hashCode() ==> 0 (every integer has the
same hashcode
too!)
> (4) integerObject.equals(integer) ==> true if same object,
false
otherwise
> By calling getInteger() from time time, getting several such
> integerObjects, can you decide just from calls to the above
API which
> of them are even and which arenÕt? No, the best you can do
is make
> arbitrary decisions, which wouldnÕt be consistent from one
run of the
> demo to another.
An integer has an integer value. If you have
integerObject.add(IntegerObject i)
returning the sum, then you should be able to tell which are
even.
> Can you de\fine a predicate:
> boolean isEven()
> which is well de\fined, using *only* calls to the API \
IÕve
speci\fied
above?
> No, you canÕt. The best you can do is randomly decide for
each new
> integerObject whether itÕs even or odd, and keep a cache \
of
such
> decisions so you donÕt contradict yourself later, but
thatÕs not a
> de\finition, thatÕs a random sampler, and \
again, just like
the manual
> demo, youÕd get different results with each run of the
program, so your
> function/method doesnÕt *de\fine* a function, \
it merely
generates a new
> random sample each time itÕs run.
ThatÕs irrelevant. Half of the integers are even.
> By the way, with *all* the integers, not just the positive
integers,
> even if you know the total ordering, thatÕs still not
enough to de\fine
> what is even and what isnÕt even. The best you can do is
divide the
> ordered set into two subsets, alternating, and then pick
one at random
> to be even. And itÕs even worse: If you have a playpen
where you can
> call an API which gives you random integers and tells for
any two
> integers whether they are equal and if not which is the
smaller of the
> two, since you donÕt know whether two integers are \
adjacent
or not you
> canÕt in a \finite number of steps determine \
the two
alternating sets.
> The set of all sets ...
> ThereÕs no such thing.
No, there is. Obviously enough there is not in ZF.
> If you understood proof-by-contradiction I could
> present you a very simple proof to that fact, but you \
donÕt
seem to
> understand even that simple aspect of mathematical logic so
it would be
> a waste of my time to present it to you.
No, youÕre wrong. Several theories including my own have \
sets
of all
sets.
> half of the integers are even.
> As just a set, 99% of them are even too.
No, the integers have integer values.
> Bijections exist between the integers and even integers,
and the
> particular one f(x)=2x, a plain straight line, shows that
there are
> twice as many integers as even integers, in the integers or
superset of
> the integers.
> It shows no such thing!! The bijection shows there are
exactly the same
> number of integers as even integers. For every integer
thereÕs a
> corresponding even integer, and vice versa.
YouÕre wrong, it shows exactly that thing.
> What you said is equivalent to saying thereÕs a bijection
between the
> \fingers on my left hand and the \fingers on my \
right hahd,
which shows
> there are twice as many \fingers on my left hand as on my
right hand.
No, it doesnÕt.
Look at any function from the integers to the integers of the
form y=mx+b,
a
straight line, for integer m. The range has asymptotic
density of 1/m in
the
integers. As it is so for any straight line function, except
for arguably
m=0,
where that bijection exists for the domain of the integers,
it shows that
the
range has an asyptotic density, which is a useful comparison
of setÕs sizes,
in
this case the sets of the domain and range, comparing the
integers to a
subset of
the integers and illustrating why the subset comprises half
of the integers,
or
generally 1/m.
Do you not see how terribly, horribly wrong youÕve been \
about
all this?
Half of the integers are even, true or false? ItÕs true. If
itÕs false
then
through contradiction the integer is neither even nor odd.
Besides your
caffeinated integers, an integer is even or odd.
Anyways, the key point to consider is that the proper subset
de\finition of
sizing
is universally applicable. Also, when you talk about sets of
only integers,
not
labels but integers, then all structural aspects of the
integers hold true
in the
comparison of collections of them.
Ross
===
Subject: Re: Proposed de\finition for comparing the sizes of
two sets
> Proposed De\finition: A set Y is said to be larger than a set
X iff there
> exists no function mapping X onto all of Y. (i.e. there is
no surjection
> from X to Y)
This is no different that what is already being used.
Because magnitudes of sets are linear ordered
X < Y iff not Y <= X
(you may prefer to read X < Y as |X| < |Y|)
YouÕve presented
X < Y when for all f:X -> Y, f not a surjection
Thus not X < Y iff some f:X -> Y, f is a surjection
which by above is equivalent to Y <= X and the de\finition
or theorem
Y <= X when some surjection f:X -> Y
is well known along with itÕs intimate associate AxC.
> Is this a workable de\finition that covers all cases?
> If so, is it widely used?
Yes and Yes.
===
Subject: Re: Proposed de\finition for comparing the sizes of
two sets
>> Proposed De\finition: A set Y is said to be larger than a
set X iff there
>> exists no function mapping X onto all of Y. (i.e. there is
no surjection
>> from X to Y)
>This is no different that what is already being used.
>Because magnitudes of sets are linear ordered
> X < Y iff not Y <= X
>(you may prefer to read X < Y as |X| < |Y|)
Linear ordering of the magnitudes is equivalent to the
axiom of choice. There are lots of models without
it, and the de\finition is never a good one without it.
--
This address is for information only. I do not claim that
these views
are those of the Statistics Department or of Purdue
University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
===
Subject: Re: So you want to count an in\finite power set ?
> To construct a powerset of an in\finite set I use an
> in\finite cartesian grid populated with 1s and 0s,
When you say in\finite cartesian grid, do you mean a
two-dimensional
grid, consisting of elements indexed by the positive integers
along
each axis? If so, thatÕs not enough elements to hold the
powerset.
> then use logical AND operation with those values from every
row
> applied to the original sequence.
What original sequence?? You need to start at the beginning.
De\fine
what you have to start with. De\fine all the personal jargon
you use.
De\fine what you construct at each step of the way. If you jump
in the
middle where you somehow already have some original sequence,
but
donÕt tell us where that original sequence came from or what
it is, we
canÕt possibly be expected to understand what \
youÕre talking
about.
> SET = { <0,3>, <1,1>, <2,4>, <4,1>, <5,5> ... }
I see no pattern there, so I have no idea what ... means at
the end.
Please de\fine clearly what SET is supposed to contain.
I donÕt even know what you mean by each individual element,
such as
<0,3>. Is that supposed to be an ordered pair, or what?
> P_1(SET) = {
> {1000000000000.. AND SET},
> {0100000000000.. AND SET},
> {1100000000000.. AND SET}...}
I assume each of the things that looks like 1000000000000..
etc. is
supposed to be a bitvector that is in\finitely long, right? I
have no
idea how to perform an AND operation between a bit vector and
a set of
ordered pairs. Please de\fine what AND does, or tell how SET is
supposed
to be treated as if it were a bit vector so that AND would
apply in the
usual way as the bitwise-AND operation.
===
Subject: Re: So you want to count an in\finite power set ?
posting-account=Qiuj5AwAAACmGnmS12qcvqA9IXzD0s4L
> P_1(SET) = {
> {1000000000000.. AND SET},
> {0100000000000.. AND SET},
> {1100000000000.. AND SET}...}
That is my only introduced notation, but I manually calculate
the step
straight after.
P_1(SET) = {
{1 AND <0,3>, 0 AND <1,1>, 0 AND <2,4> ...},
{0 AND <0,3>, 1 AND <1,1>, 0 AND <2,4> ...},
{1 AND <0,3>, 1 AND <1,1>, 0 AND <2,4> ...},
...}
1 = true, 0 = false. TRUE AND X = X
Look up Ôbit masking
The original SET is { <0,3>, <1,1>, <2,4>, ...}
i.e. pi 314159.. put into <0 > <1 > <2 > <3 > ...
Its the quickest way I could think of to make in\finite
distinct members
that werent trivial N.
I use { } for set and < > for sequence, fairly standard for
programmers
atleast.
Herc
===
Subject: Mathematical Ideas
I am trying to \find an idea in the area of number theory or
combinatorics that I can do my undergraduate senior research
paper/presentation on. IÕve seen how people on here have
harrassed
others asking for idea telling them to go see their advisor.
However I have a good case for asking here: the two math
profs at my
school only have interest in statistics, calculus, and
geometry. And
these are not my strong \fields, as I am double majoring in
math and
computer science. I have met with my professors once a week
on average
for the entire semester and together have still to come up
with a
topic they deem viable. I have to do more than just research
the
idea, but in some way I have to do something with it or
contribute to
the topic (nothing major mind you). So if you have any
constructive
===
Subject: connecting opposite edges of the cube
If the 3-cube is a \finite union {A_i} of open subsets,
any 3 of them having empty intersection, is it true that
some A_i is intersecting two opposite edges of the cube?
Opposite here means symmetric with respect to the
cubocenter.
[I proved that if the 3-cube is a \finite union {A_i} of
open subsets, any 4 of them with empty intersection,
then some A_i is intersecting two opposite FACES of the
cube]
vedran
===
Subject: Philosopher/marketer is stumped by this Probability
Problem. Plz
help.
Hi. I hope someone can at least point me in the right
direction.
IÕm looking for a general solution to the following problem:
Suppose you have two urns, each containing an equivalent VERY
LARGE
(maybe not in\finite, but very large) number of balls. Each urn
has
only red and green balls in it.
Suppose you draw from urn A 11 times, with replacement, and
you get 7
green balls. So the ratio of green balls to total draws in
the sample
is 7/11.
Suppose you draw from urn B 15 times, with replacement, and
you get 8
green balls. So the ratio, for urn B, of green balls to total
draws
in the sample is 8/15.
Assuming random sampling, and given these samples, what is the
probability that urn A contains more green balls than urn B?
I am actually looking for the general solution. Given a
sample from
urn A of X1/Y1, and a sample from urn B of X2/Y2 (all Xs and
Ys are
integers), what is the probability that there are more greens
in urn A
than in urn B.
IÕm kind of stumped.
Giblar
===
Subject: Re: Philosopher/marketer is stumped by this
Probability Problem.
Plz help.
>Hi. I hope someone can at least point me in the right
direction.
>IÕm looking for a general solution to the following \
problem:
>Suppose you have two urns, each containing an equivalent
VERY LARGE
>(maybe not in\finite, but very large) number of balls. Each
urn has
>only red and green balls in it.
>Suppose you draw from urn A 11 times, with replacement, and
you get 7
>green balls. So the ratio of green balls to total draws in
the sample
>is 7/11.
>Suppose you draw from urn B 15 times, with replacement, and
you get 8
>green balls. So the ratio, for urn B, of green balls to
total draws
>in the sample is 8/15.
>Assuming random sampling, and given these samples, what is
the
>probability that urn A contains more green balls than urn B?
>I am actually looking for the general solution. Given a
sample from
>urn A of X1/Y1, and a sample from urn B of X2/Y2 (all Xs and
Ys are
>integers), what is the probability that there are more
greens in urn A
>than in urn B.
>IÕm kind of stumped.
There is not much information on the problem. What does
equivalent mean?
If the total number of balls is equal, the is the standard
problem of comparing proportions. One still needs prior
assumptions to compute the probability.
--
This address is for information only. I do not claim that
these views
are those of the Statistics Department or of Purdue
University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
===
Subject: Re: Philosopher/marketer is stumped by this
Probability Problem.
Plz help.
>IÕm looking for a general solution to the following \
problem:
>Suppose you have two urns, each containing an equivalent
VERY LARGE
>(maybe not in\finite, but very large) number of balls. Each
urn has
>only red and green balls in it.
>Suppose you draw from urn A 11 times, with replacement, and
you get 7
>green balls. So the ratio of green balls to total draws in
the sample
>is 7/11.
>Suppose you draw from urn B 15 times, with replacement, and
you get 8
>green balls. So the ratio, for urn B, of green balls to
total draws
>in the sample is 8/15.
>Assuming random sampling, and given these samples, what is
the
>probability that urn A contains more green balls than urn B?
>I am actually looking for the general solution. Given a
sample from
>urn A of X1/Y1, and a sample from urn B of X2/Y2 (all Xs and
Ys are
>integers), what is the probability that there are more
greens in urn A
>than in urn B.
If you actually want a *probability*, you need to specify a
prior joint
distribution on the fractions of green balls in the two urns.
Use
Bayesian updating.
It is also possible to do a frequentist hypothesis test: Y1
and Y2
should be slightly larger (but not necessarily too much so),
and use the
test statistic (Q1-Q2) / sqrt(Qc (1-Qc) / (Y1+Y2)), which has
approximately a standard normal distribution. Here, Qi = Xi /
Yi,
and Qc = (X1+X2) / (Y1+Y2)
--
Stephen J. Herschkorn sjherschko@netscape.net
===
Subject: Re: Philosopher/marketer is stumped by this
Probability Problem.
Plz help.
[bccÕd to OP, who e-mailed me]
You asked what is the *probability* that one method is better
than the
other, but I suspect that that is not what you really want.
In the
frequentist paradigm, one poses hypotheses in terms of unknown
parameters which are not random variables, hence it makes no
sense to
speak of the probability of a hypothesis. Rather, one decides
whether
an observed event is excessively improbable in the case that a
hypothesis is true; if so, one rejects the hypothesis and
accepts the
alternative.
Suppose you want to ask the question, Is there evidence that
two
methods differ in their ef\ficacy? The frequentist method is as
follows. Let p1 and p2 be the yields of the two methods. You
pose
the null hypothesis and its alternative,
H0: p1 = p2
H1: p1 != p2
Under the null hypothesis, the test statistic Z = (Q1-Q2) /
sqrt(Qc
(1-Qc) / n) has approximately a standard normal distribution,
where the
QÕs are de\fined as in my post and n is the \
combined sample
size (your
Y1+Y2). You want Y1 and Y2 to be suf\ficiently large; if each
are at least 15, you are probably O.K. Thus, if the observed
value of
the test statistic is too extreme (e.g., if P{|Z| > z_obs} <
0.10;
0.10 is called ths signi\ficance level), you reject the null
hypothesis
and \find evidence of a difference. Ohterwise, you \
\find no
evidence of a
difference. This is all rather standard stuff; also, it makes
several
assumptions, in particular, independent samples.
In Bayesian statistics, one models the unknown parameters as
random
variables, hence one can come up with a coherent statement of
the
probabiliy of a difference given the data. This requires the
speci\fication of a prior joint distribution on the parameters,
and one
computes the posterior conditional probability given the
observed data.
Standard elementary texts on statistics should discuss all of
this.
>> IÕm looking for a general solution to the following
problem:
>> Suppose you have two urns, each containing an equivalent
VERY LARGE
>> (maybe not in\finite, but very large) number of balls. Each
urn has
>> only red and green balls in it.
>> Suppose you draw from urn A 11 times, with replacement,
and you get 7
>> green balls. So the ratio of green balls to total draws in
the sample
>> is 7/11.
>> Suppose you draw from urn B 15 times, with replacement,
and you get 8
>> green balls. So the ratio, for urn B, of green balls to
total draws
>> in the sample is 8/15.
>> Assuming random sampling, and given these samples, what is
the
>> probability that urn A contains more green balls than urn
B?
>> I am actually looking for the general solution. Given a
sample from
>> urn A of X1/Y1, and a sample from urn B of X2/Y2 (all Xs
and Ys are
>> integers), what is the probability that there are more
greens in urn A
>> than in urn B.
>>
> If you actually want a *probability*, you need to specify a
prior
> joint distribution on the fractions of green balls in the
two urns.
> Use Bayesian updating.
> It is also possible to do a frequentist hypothesis test: Y1
and Y2
> should be slightly larger (but not necessarily too much
so), and use
> the test statistic (Q1-Q2) / sqrt(Qc (1-Qc) / (Y1+Y2)),
which has
> approximately a standard normal distribution. Here, Qi = Xi
/ Yi,
> and Qc = (X1+X2) / (Y1+Y2)
--
Stephen J. Herschkorn sjherschko@netscape.net
===
Subject: Re: Philosopher/marketer is stumped by this
Probability Problem.
Plz help.
Hi Stephen (Keith and Herman, too!)
might be able to use much of your present answers, but I
worry now
that my Urn analogy isnÕt apt for my real problem.
Please, if youÕll indulge me, let me state my real-world
application,
then maybe you can help me set up the problem correctly.
(In other words, IÕm not really dealing with urns \
\filled with
red
and green balls :-) )
IÕm developing a marketing tool that analyzes split run
tests. So,
say, on a sales letter, copies that run headline A might have
a sales
conversion ratio of 5 sales out of 347 exposures, and copies
that run
headline B might have a sales conversion ratio of 6 sales out
of 334
exposures. I want to provide an estimate of how likely it is
that
ultimately headline B will convert better than headline A.
This will
help determine whether to continue running the test or to
terminate
it.
I thought the urn problem was a good analogy, but it seems
that I
wasnÕt explicit about several assumptions, and those
assumptions were
important for answering the question.
If you have a straightforward solution, please ignore the
rest of this
message and proceed to enlighten me. I would be much grateful.
But to show IÕm not just being lazy, and that \
IÕm trying to
\figure it
out, here is an approach I came up with, but have minimal
con\fidence
in.
One approach I thought might work would be to \figure out some
intermediate value X between the two conversion ratios (i.e.,
in this
case 5/347 < X < 6/334)where
Prob(CRA>=X | (EA=347) & (SA<=5)) = Prob( CRB<=X |(EB=334) &
(SB>=6))
Where CRA = the true conversion rate of letter A
CRB = the true conversion rate of letter B
SA = Sales with letter A
SB = Sales with letter B
EA = Exposures for letter A
EB = Exposures for letter B
Then, it seems that the probability that letter B does NOT
ultimately
have a higher conversion rate than letter A is the
probability of a
conjunction, namely,
Prob[(SA<=5 | (EA=347) & (CRA=X)) & (SB>=6 | (EB=334) &
(CRB=X))]
Call this Z.
And then the probability that letter B has a greater
conversion ratio
than letter A is 1-Z.
But this seems a lot more complicated than IÕm hoping it has
to be.
And IÕm not sure itÕs right anyway. And, of \
course, again,
IÕm after
the general solution.
IÕm sorry if you wind up repeating yourself, but I wanted to
make sure
you had more information before I tried to apply the advice
already
given.
Giblar. (By the way, I abandoned mathematics in college to
pursue
philosophy. So far it seems the mathematics would have been
more
useful :-)
>IÕm looking for a general solution to the following \
problem:
>Suppose you have two urns, each containing an equivalent
VERY LARGE
>(maybe not in\finite, but very large) number of balls. Each
urn has
>only red and green balls in it.
>Suppose you draw from urn A 11 times, with replacement, and
you get 7
>green balls. So the ratio of green balls to total draws in
the sample
>is 7/11.
>Suppose you draw from urn B 15 times, with replacement, and
you get 8
>green balls. So the ratio, for urn B, of green balls to
total draws
>in the sample is 8/15.
>Assuming random sampling, and given these samples, what is
the
>probability that urn A contains more green balls than urn B?
>I am actually looking for the general solution. Given a
sample from
>urn A of X1/Y1, and a sample from urn B of X2/Y2 (all Xs and
Ys are
>integers), what is the probability that there are more
greens in urn A
>than in urn B.
>
> If you actually want a *probability*, you need to specify a
prior joint
> distribution on the fractions of green balls in the two
urns. Use
> Bayesian updating.
> It is also possible to do a frequentist hypothesis test: Y1
and Y2
> should be slightly larger (but not necessarily too much
so), and use the
> test statistic (Q1-Q2) / sqrt(Qc (1-Qc) / (Y1+Y2)), which
has
> approximately a standard normal distribution. Here, Qi = Xi
/ Yi,
> and Qc = (X1+X2) / (Y1+Y2)
===
Subject: Re: Philosopher/marketer is stumped by this
Probability Problem.
Plz help.
>Suppose you have two urns, each containing an equivalent
VERY LARGE
>(maybe not in\finite, but very large) number of balls. Each
urn has
>only red and green balls in it.
>Suppose you draw from urn A 11 times, with replacement, and
you get 7
>green balls. So the ratio of green balls to total draws in
the sample
>is 7/11.
>Suppose you draw from urn B 15 times, with replacement, and
you get 8
>green balls. So the ratio, for urn B, of green balls to
total draws
>in the sample is 8/15.
>Assuming random sampling, and given these samples, what is
the
>probability that urn A contains more green balls than urn B?
Since nobody else has said anything, IÕll start on this
problem and see
where it goes. No promises...
I think for your purposes you can simplify this from the
discrete problem
of
ball ratios to a more abstract one involving real number
probabilities.
Let green_A be the ratio of green balls to total balls in urn
A, and
green_B
be the same ratio in urn B.
Your samples are cases of binomial distribution.
P(7 of 11 from A) = choose(11,7) * green_A^7 * (1-green_A)^4
P(8 of 15 from B) = choose(15,8) * green_B^8 * (1-green_B)^7
choose(n,m) = n! / (m! * (n-m)!)
P(7 of 11 from A) = 330 * green_A^7 * (1-green_A)^4
P(8 of 15 from B) = 5435 * green_B^8 * (1-green_B)^7
To continue, we need to assume a probability distribution for
green_A and
green_B. LetÕs assume a uniform distribution; that is, \
1/1000
green is
just
as likely as 500/1000. We are going to work in the samples
next, so donÕt
worry about that. I assume you havenÕt been given any other
info about
what
the distribution might be, such as at least 25% of the balls
in each urn
are green.
S = sample event
R = ratio event
P(S&R) = P(R) * P(S|R)
P(S1&R1&S2&R2) = P(R1) * P(S1|R1) * P(R2) * P(S2|R2)
We know both P(S|R) from the binomial distributions, and
weÕre assuming
P(R)
to be uniform on [0,1] so I think that:
P(green_A > green_B) =
int(green_A=0,1)[int(green_B=0,green_A)[
330 * green_A^7 * (1-green_A)^4 * 5435 * green_B^8 *
(1-green_B)^7
* 1 * 1]]
A (double) polynomial integral -- not too hard but the more
samples you
have
the more terms you get.
A good check would be to reverse the samples, and the two
probabilities
should add to 1. If I made a mistake they probably wonÕt.
--Keith Lewis klewis {at} mitre.org
The above may not (yet) represent the opinions of my employer.
===
Subject: Continued Fraction of n*x with n-th Partial Quotient
= 1
Consider a real number x that satis\fies the condition:
the n-th partial quotient of
the continued fraction expansion of n*x
equals 1 for all positive integer n
where the constant term is assigned index 1.
Example.
A possible solution to x such that CF(n*x)[n]=1 may begin:
x=1.8661698591292536563675165915962758366350388262...
and the continued fraction expansions of n*x are
CF(x)=[_1; 1, 6, 2, 8, 2, 11, 1, 1, 1, 1, 22,...]
CF(2x)=[3;_1, 2, 1, 2, 1, 3, 1, 2, 1, 5, ...]
CF(3x)=[5; 1,_1, 2, 26, 2, 3, 1, 1, 6, 1,...]
CF(4x)=[7; 2, 6,_1, 1, 2, 1, 2, 2, 1, 1,...]
CF(5x)=[9; 3, 44, 2,_1, 1, 11, 1, 1, 4,...]
CF(6x)=[11; 5, 13, 4, 1,_1, 3, 3, 2, 3, ...]
CF(7x)=[13; 15, 1, 4, 1, 2,_1, 2, 1, 2,...]
CF(8x)=[14; 1, 13, 6, 2, 2, 4,_1, 2, 1,...]
CF(9x)=[16; 1, 3, 1, 8, 6, 1, 5,_1, 1, ...]
CF(10x)=[18; 1, 1, 1, 21, 1, 2, 3, 5,_1, ...]
where the partial quotients along the main diagonal are all
to equal 1.
It seems that there may be an in\finite number
of attractors that satisfy this condition.
What I am interested in are the extremes
(assuming that at least one solution exists).
The minimum value for such an x is greater than 1.25,
and the maximum value of such x is less than 1.88.
What are the aproximate minimum and maximum values
that satisfy the condition?
Paul
===
Subject: Re: 4400
> The reason I posted that was to inform other people exactly
what the heck
> John was talking about.
Well, they donÕt know the Usenet Rules*:
Usenet Rule #37 (Faisal Nameer Jawdat): Read the thread from
the beginning, or else.
(* For you newbies, these rules can be found at
http://www.faqs.org/faqs/usenet/legends/godwin/ at the bottom
of the page.)
-- Christopher Heckman, Usenet surfer since 1994.
> Similarly to his post, I should have given more information.
>
> American Zoetrope, comes a haunting new limited series: THE
4400.
> Over the last century, thousands of people have gone
missing.
> Suddenly
> and inexplicably, 4400 missing people are returned all at
once, as
they
> were
> on the day they vanished. Unclear what this world
altering-event
means,
> the
> government investigates the 4400 to piece together where
theyÕve been
> and
> why theyÕve been returned. It quickly becomes apparent \
that
their
> presence
> will change the human race in ways no one could have ever
foreseen.
>
> post: Why 4400, instead of, say, 5280?
> -- Christopher Heckman
> Can anyone tell me what is special about the number 4400?
> As in the TV show the the 4400
> what does it stand for?
> Its not a prime number nor does it have a even square root.
> any ideas???
>
===
Subject: Re: 4400
Sorry my friend, but the information I posted is no where in
this thread -
hence the reason why I posted it. But IÕm sure many other
people will
listen to your wisdom for future posts.
> The reason I posted that was to inform other people exactly
what the
heck
> John was talking about.
> Well, they donÕt know the Usenet Rules*:
> Usenet Rule #37 (Faisal Nameer Jawdat): Read the thread from
> the beginning, or else.
> (* For you newbies, these rules can be found at
> http://www.faqs.org/faqs/usenet/legends/godwin/ at the
bottom of the
page.)
> -- Christopher Heckman, Usenet surfer since 1994.
> Similarly to his post, I should have given more information.
>
company,
> American Zoetrope, comes a haunting new limited series: THE
4400.
> Over the last century, thousands of people have gone
missing.
> Suddenly
> and inexplicably, 4400 missing people are returned all at
once, as
they
> were
> on the day they vanished. Unclear what this world
altering-event
means,
> the
> government investigates the 4400 to piece together where
theyÕve
been
> and
> why theyÕve been returned. It quickly becomes apparent \
that
their
> presence
> will change the human race in ways no one could have ever
foreseen.
>
> post: Why 4400, instead of, say, 5280?
> -- Christopher Heckman
> Can anyone tell me what is special about the number 4400?
> As in the TV show the the 4400
> what does it stand for?
> Its not a prime number nor does it have a even square root.
> any ideas???
>
===
Subject: Re: 4400
| Can anyone tell me what is special about the number 4400?
|
| As in the TV show the the 4400
| what does it stand for?
| Its not a prime number nor does it have a even square root.
|
| any ideas???
It has a square root which I can express precisely in the
form of a
repeating pattern in the Stern-Brocot number system. Start
with a
Stern-Brocot tree. Descend 66 steps to the right, followed by
3 steps
to the left, followed by 66 more steps to the right. Repeat
the 66R,
3L, 66R sequence over and over and it converges on the square
root of
4400.
That square root can be approximated by the following
fraction of whole
numbers:
835119505470355596931039942986758467 /
12589900248874196950608348266938385
Also:
If you take an ASCII code value for a character used to
represent a digit
in the hexadecimal number system, including either upper or
lower case
for those digits which are normally letters (e.g. a..f and
A..F), an do
a bitwise logical-OR of those code with the bits representing
the value
4400, then what you get is a number that when divided by 55
results in the
value that ASCII character represents in the remainder of
that division.
--
-------------------------------------------------------------
---------------
-
| Phil Howard KA9WGN | http://linuxhomepage.com/
http://ham.org/
|
| (\first name) at ipal.net | http://phil.ipal.org/
http://ka9wgn.ham.org/
|
-------------------------------------------------------------
---------------
-
===
Subject: Re: logic of the Cantorian followers mind
> IÕve been told CantorÕs proof has nothing to \
do with
computability.
Diagonalization is a powerful technique with many
applications. It can be
used to show, among other things, that there are uncomputable
functions
from N -> N.
> I donÕt \find it strange that I was able to \
lead you here
though.
The concept of computability relates to the existence of
unidenti\fiable
real numbers insofar as there is to be an effective method for
distinguishing between identi\fiers.
===
Subject: Re: 2-manifold metric spaces with many symmetries
> A property of euclidean 2-D space is that:
> Given two congruent triangles with distinct edge-lengths,
say
> triangle(A, B, C) and triangle(AÕ, BÕ, \
CÕ) , then there is
just one
> isometry that maps the \first triangle to the second. ( A, B,
C, AÕ,
> BÕ, Care points in euclidean 2-D space.) [ \
Property 1 ]
> I think the same is true for a torus, viewed as a
\finite-height
> cylinder with vertical axis of rotation and where the upper
boundary
> of the cylinder is glued to the lower boundary.
>> The result does not hold for the torus T^2. T^2 has all the
>> translation symmetries of R^2, but rotations in T^2 (i.e.,
>> isometries of T^2 that \fix a point) are severely limited.
> [...]
> That came as a surprise to me. But now I see that viewing
T^2
[...]
>> Presumably, you want the surface to be a smooth submanifold
>> of R^3.
> I want the surface to be a smooth submanifold of R^n,
> for some positive n.
> ( I had a look at Riemannian manifold
> here:
> http://en.wikipedia.org/wiki/Riemannian_manifold )
For more on the geometry of spacetime assuming
spatial homogeneity and isotropy: (perhaps this is physics?)
List of links:
David EppsteinÕs Geometry Junkyard page:
http://www.ics.uci.edu/~eppstein/junkyard/topic.html
has a Hyperbolic Geometry link
which has this link: Visualization of a hyperbolic universe,
(by Martin Bucher)
which takes one here: (Martin BucherÕs homepage):
http://www.damtp.cam.ac.uk/user/mab43/
There one \finds images for a hyperbolic (3-D) universe,
a ßat or euclidean universe and a spherical universe.
Also, the following text:
The above three images illustrate the difference
in perspective of the three types spacetime
geometries possible if the requirements of
spatial homogeneity and isotropy are imposed.
Perhaps this is a result in the domain of physics
(General Relativity). Anyway, the images are interesting.
David Bernier
===
Subject: Re: 2-manifold metric spaces with many symmetries
>
>> A property of euclidean 2-D space is that:
>>
>> Given two congruent triangles with distinct edge-lengths,
say
>> triangle(A, B, C) and triangle(AÕ, BÕ, \
CÕ) , then there is
just
>> one isometry that maps the \first triangle to the second. (
A,
>> B, C, AÕ, BÕ, Care points \
in euclidean 2-D space.)
>> [ Property 1 ]
>>
>> I think the same is true for a torus, viewed as a
\finite-height
>> cylinder with vertical axis of rotation and where the upper
>> boundary of the cylinder is glued to the lower boundary.
>>
>
> The result does not hold for the torus T^2. T^2 has all the
> translation symmetries of R^2, but rotations in T^2 (i.e.,
> isometries of T^2 that \fix a point) are severely limited.
>> [...]
>> That came as a surprise to me. But now I see that viewing
T^2
> [...]
> Presumably, you want the surface to be a smooth submanifold
of
> R^3.
>> I want the surface to be a smooth submanifold of R^n, for
some
>> positive n.
>> ( I had a look at Riemannian manifold here:
>> http://en.wikipedia.org/wiki/Riemannian_manifold )
> For more on the geometry of spacetime assuming spatial
homogeneity
> and isotropy: (perhaps this is physics?)
> List of links:
> David EppsteinÕs Geometry Junkyard page:
> http://www.ics.uci.edu/~eppstein/junkyard/topic.html
> has a Hyperbolic Geometry link which has this link:
Visualization
> of a hyperbolic universe, (by Martin Bucher)
> which takes one here: (Martin BucherÕs homepage):
> http://www.damtp.cam.ac.uk/user/mab43/
> There one \finds images for a hyperbolic (3-D) universe, a
ßat or
> euclidean universe and a spherical universe.
> Also, the following text:
> The above three images illustrate the difference in
perspective of
> the three types spacetime geometries possible if the
requirements of
> spatial homogeneity and isotropy are imposed.
> Perhaps this is a result in the domain of physics (General
> Relativity). Anyway, the images are interesting.
Probably not. Physics is not geometry. ItÕs more like
differential
geometry.
> David Bernier
The result you want requires the manifold to have a transitive
action of isometries by some Lie group G; in addition, this
action must contain the rotation group SO(n) in the isotropy
subgroup of G at x for each element x. That means that your
n-manifold has a symmetry group of dimension at least
n + n(n-1)/2 = n(n+1)/2. I note that this is the dimension
of the group SO(n+1), the group of isometries of the n-sphere.
IÕll note that among the various differential structures on
S^n (for n >= 7, these are non-diffeomorphic smooth structures
on the same topological space S^n), the only one that has a
group of isometries of this dimension is the standard, round
sphere.
I suspect that this condition poses a severe constraint on
the manifold, since itÕs as symmetric as the n-sphere.
Dale
===
Subject: Re: 2-manifold metric spaces with many symmetries
Yet another correction. I should have proofread this
stuff better.
... stuff deleted ...
> I want the surface to be a smooth submanifold of R^n, for
some
> positive n.
... more stuff deleted ...
Here, I hadnÕt scanned up to see the use of n as the \
dimension
of the ambient space, and used it incorrectly as the
dimension of
the submanifold; IÕve replaced n by k in this paragraph, and
note
that it refers to the dimension of the submanifold:
> The result you want requires the manifold to have a
transitive
> action of isometries by some Lie group G; in addition, this
> action must contain the rotation group SO(k) in the isotropy
> subgroup of G at x for each element x. That means that your
> k-manifold has a symmetry group of dimension at least
> k + k(k-1)/2 = k(k+1)/2. I note that this is the dimension
> of the group SO(k+1), the group of isometries of the
k-sphere.
> IÕll note that among the various differential structures \
on
> S^k (for k >= 7, these are non-diffeomorphic smooth
structures
> on the same topological space S^k), the only one that has a
> group of isometries of this dimension is the standard, round
> sphere.
> I suspect that this condition poses a severe constraint on
> the manifold, since itÕs as symmetric as the k-sphere.
> Dale
There it is. Now get back to work.
Dale.
===
Subject: Re: 2-manifold metric spaces with many symmetries
Just \fixing a poorly-worded sentence.
... stuff deleted ...
>> Perhaps this is a result in the domain of physics (General
>> Relativity). Anyway, the images are interesting.
> Probably not. Physics is not geometry. ItÕs more like
differential
> geometry.
What I meant to have said was that the OPÕs problem is more
like
differential geometry than it is like physics.
>> David Bernier
... the rest deleted ...
> Dale
===
Subject: Re: mean value of the roots of a stochastic polynom
a is \fixed, and the hypothesis b << c wonÕt be \
too false, so
that the
series expansion migth be an interesting solution.
I will try it for the case where b is follows a Weibull
distribution
and where the pdf of c is empirically known.
Manu
===
Subject: WhatÕs this function called?
I have de\fined the following functions, which I \
\find intriguing
and
beautiful. I vaguely remember seeing some of it before, so I
doubt its
that original. Can anyone tell me what these functions called?
For all integers n, we de\fine s(n) as the sum of all the prime
factors
of n, counting repeated factors multiple times. For example
s(10) = 7,
since 10=2*5; 8=2*2*2, thus s(8) = 6. And s(p) = p for all
primes p.
s(mn) = s(m) + s(n).
We can extend it to negative integers, so that s(-m) = s(m) +
1/2
Argument: s(mn) = s(m) + s(n), s(-1 * -1) = s(-1) + s(-1) =
s(1), s(1)
= 1, hence 2*s(-1) = 1, hence s(-1) = 1/2. s(-m) = s(-1 * m)
= s(-1) +
s(m),
thus s(-m) = s(m) + 1/2.
Also, s(m^n) = n*s(m), so arguably s(m^-1) = -s(m). Thus
s(m/n) = s(m)
- s(n). Which provides an extension to the rationals. [Could
it
sensibly be extended further, say to the reals?]
Can we give a sensible value to s(0)? Say, s(0) = 0?
But, ignoring this extension to negative integers and
rationals: Now,
we can iterate this function s, by applying its result to
itself. For
example s(8) = 6. s(6) = 5. Let us de\fine another function
t(n), to be
the result of iterating s(n) until we keep on getting the
same value
(which will be a prime, or the number 4).
Finally, we can de\fine another function u(n), which is the
number of
times we must iterate s(n) before the process terminates.
For example u(4) = 0, since s(4) = 4.
u(any prime) = 0, since s(any prime) = that prime.
u(6) = 1, since s(6) = 5, and s(5) = 5.
u(8) = 2, since s(8) = 6, s(6) = 5, and s(5) = 5.
So basically, are there names for these functions I have
termed s, t,
u?
Simon Kissane
===
Subject: Re: WhatÕs this function called?
> For all integers n, we de\fine s(n) as the sum of all the
prime factors
> of n, counting repeated factors multiple times. For example
s(10) = 7,
s(0) = 0 = s(1) ?
> s(mn) = s(m) + s(n).
YouÕre messing around with a homomorphism from a
multiplicative group to
an additive group. ThatÕs all youÕre doing.
s(0) = s(0*7) = s(0) + s(7)
s(0) = s(0*5) = s(0) + s(5)
s(0) cannot be de\fined when requiring s(mn) = s(m) + s(n)
> We can extend it to negative integers, so that s(-m) = s(m)
+ 1/2
> Argument: s(mn) = s(m) + s(n), s(-1 * -1) = s(-1) + s(-1) =
s(1), s(1)
> = 1, hence 2*s(-1) = 1, hence s(-1) = 1/2. s(-m) = s(-1 *
m) = s(-1) +
> s(m),
> thus s(-m) = s(m) + 1/2.
s(1) = s(1*1) = s(1) + s(1); s(1) = 1 ??? No! s(1) = 0
So explain this wonderful mess
s(1) = s(-1 * -1 * -1 * -1) = s(-1) + s(-1) + s(-1) + s(-1)
= 4(s(1) + 1/2) = 4s(1) + 2
Or this one
s(2) = s(--2) = s(-2) + 1/2 = s(2) + 1
> Also, s(m^n) = n*s(m), so arguably s(m^-1) = -s(m). Thus
s(m/n) = s(m)
> - s(n). Which provides an extension to the rationals.
[Could it
s(1) = s((-1)^2) = 2s(-1)
s(1) = s((-1)^4) = 4s(-1)
s(-1) = 0, youÕve no other choice
> sensibly be extended further, say to the reals?]
log ab = log a + log b for a,b > 0
log 1 = 0
log a^n = n log a
log a/b = log a - log b
log |ab| = log |a| + log |b| for all the reals /= 0
> Can we give a sensible value to s(0)? Say, s(0) = 0?
No as explain above. Also
s(1) = s(-1) = 0, so s(-n) = s(-1*n) = s(-1) + s(n) = s(n)
> But, ignoring this extension to negative integers and
rationals: Now,
> we can iterate this function s, by applying its result to
itself. For
> example s(8) = 6. s(6) = 5. Let us de\fine another function
t(n), to be
> the result of iterating s(n) until we keep on getting the
same value
> (which will be a prime, or the number 4).
So now youÕre talking about orbits of a group action or
something like
that. s(1) = 0, s(0) unde\finable, t(0), t(1) \
unde\finable
> Finally, we can de\fine another function u(n), which is the
number of
> times we must iterate s(n) before the process terminates.
> For example u(4) = 0, since s(4) = 4.
> u(any prime) = 0, since s(any prime) = that prime.
> u(6) = 1, since s(6) = 5, and s(5) = 5.
> u(8) = 2, since s(8) = 6, s(6) = 5, and s(5) = 5.
As before u(1), u(0) not de\finable.
> So basically, are there names for these functions I have
termed s, t, u?
tus
===
Subject: Re: WhatÕs this function called?
S(1) = 0 because 1 doesnÕt have any prime factors.
> s(0) = s(0*7) = s(0) + s(7)
The usual trick when dealing with this problem is to realize
that 0 is
divisible by an in\finite number of copies of any prime(s) you
want, so
s(0) is the sum of them all, i.e. in\finity. Then good old
Cantor
cardinal number arithmethic shows that in\finity plus any \
\finite
number
is in\finity again, satisfying the above equation.
The same sort of thing comes up when dealing with ideals
generated by 1
or 0 and asking questions such as unique factorization.
The OPÕs attempt to de\fine S(-1) = 1/2 did no \
good at all, as
you
pointed out, because S(-1 * -1) doesnÕt equal S(-1) + (-1).
Perhaps units as factors should be ignored, as they usually
are when
dealing with factorizations. So S(-1) = 0.
As for the OPÕs question whether it can be extended to
rationals: NO,
because in a \field *all* nonzero numbers are units, so S(n)
would have
to be zero for *all* nonzero values, contradicting the
original
de\finition involving prime factors as integers.
===
Subject: Re: Ozkural was Re: Platonism
>>
> That was a pedagogical question (e.g. how do you tell what
Z is to
> somebody whoÕs never studied formal set theory, who is
*not* me if
you
> will try another sore joke), and it was a real explanatory
issue that
> I faced trouble with over a coffee table.
>>
>> Right. You were wondering if there are integers with an
in\finite
>> number of digits.
>> Indeed. His question was not
>> How do I explain to my interlocutors that each integer has
\finitely
>> many digits in its base ten representation?
>> it was
>> are there integers with an in\finite number of digits?.
> Doh! Not interlocutors, friends.
<snip: whinging disingenuous self-justi\fication peppered with
personal attacks on yours truly excised; all of value
retained>
--
Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html
Lacan, Jacques, 79, 91-92; mistakes his penis for a square
root, 88-9
Francis Wheen, _How Mumbo-Jumbo Conquered the World_
===
Subject: Need help with inverse laplace transform!!
Hi!
I could use some help with inversing a Laplace Transform. The
function
looks as follows:
exp(-v*z)*s*(1-v)
where v=sqrt(1-2/s)
and I want to perform an inverse laplace transform (s ->u)
When I performed the inversion myself, I ended up with a
modi\fied
Besselfunction*exp(u)/u, but this does not seem to be right..
I am very grateful for any ideas!!!
/Malin
*-----------------------*
www.GroupSrv.com
*-----------------------*
===
Subject: Re: Painful but inevitable resignation
>> Of course, I too can envision probable continuation of the
trail
>> into future, extrapolated from the existing trail. But I
see such
>> continuations more as the ability to predict the logical
end or
>> continuation of coherent processes that we understand.
>> The point is, real life is never completely predictible
but embedded in
>> unseen external inßuence.
> Of course, but I donÕt see why this should be a problem,
since you
> agree tht the future is not written yet.
I see the problem in lacking public awarenes. Even John L.
Bell
http://publish.uwo.ca/~jbell/
shares the widespread deterministic notion of causality. He
is otherwise
expressing my mathematical understanding quite accurately.
> Sure. I never treated future time as if it was observable.
Was there any objection?
===
Subject: Re: Painful but inevitable resignation
>> Of course, I too can envision probable continuation of the
trail
>> into future, extrapolated from the existing trail. But I
see such
>> continuations more as the ability to predict the logical
end or
>> continuation of coherent processes that we understand.
> The point is, real life is never completely predictible but
embedded
in
>> unseen external inßuence.
> Of course, but I donÕt see why this should be a problem,
since you
> agree tht the future is not written yet.
> I see the problem in lacking public awarenes. Even John L.
Bell
> http://publish.uwo.ca/~jbell/
> shares the widespread deterministic notion of causality. He
is
> otherwise expressing my mathematical understanding quite
accurately.
I agree. I observed no awareness at all, even in the \
scienti\fic
community.
> Sure. I never treated future time as if it was observable.
> Was there any objection?
If there were, they kept their objections to themselves.
Besides,
physicists tend to dislike discussing with me. Causality is
not
very popular these days.
Andr.8e Michaud
===
Subject: High school factorization of algebraic fraction
Hi Group,
I was wondering how I could go about writing this algebraic
fraction in the
simplest factored form:
((6a+2a)/(4a+8))*((6a+12)/(a))
What steps should I take?
James Midolo
===
Subject: Re: High school factorization of algebraic fraction
> I was wondering how I could go about writing this algebraic
fraction
> in the simplest factored form:
> ((6a2+2a)/(4a+8))*((6a+12)/(a))
> What steps should I take?
When you say 6a2, do you mean 6 * a^2 ?
IÕll assume the answer is yes.
Well, before posting you should at least *try* to do the
obvious stuff.
For example, do you see that 4a+8 has a common factor of 4?
So you can
factor that as 4 * (a+2), right? So why didnÕt you at least
do that
tiny bit of the work before posting? There are two other
places where
you should have seen common factors that are obvious and easy
to pull
out. So why didnÕt you do those simple operations before
posting? Are
you so terribly shy and afraid of making the slighest mistake
and being
embarrassed that you would rather have somebody else do your
homework
for you then even try to do it yourself?? Or are you just
plain too
lazy to do your own work?
Unfortunately Jeroen Boschma already posted a response with
most of
that easy obvious stuff already done for you, and I presume
you looked
at and copied his work, so you have forever lost the chance
to \figure
that sort of stuff out yourself. On your \final exam you \
wonÕt
have him
to do your work for you, so I suggest you \find another example
from
your book that looks about as dif\ficult as this example, and
try that
example without any help, get as far along with the easy
stuff as you
can, then post the question and your partial work here and
ask if
thereÕs anything further you can do. ThatÕs \
the only way
youÕll learn
how to do these sorts of factorizations and simpli\fications of
polymomial fractions.
===
Subject: Re: High school factorization of algebraic fraction
> I was wondering how I could go about writing this algebraic
fraction
> in the simplest factored form:
> ((6a2+2a)/(4a+8))*((6a+12)/(a))
> What steps should I take?
> When you say 6a2, do you mean 6 * a^2 ?
> IÕll assume the answer is yes.
> Well, before posting you should at least *try* to do the
obvious stuff.
> For example, do you see that 4a+8 has a common factor of 4?
So you can
> factor that as 4 * (a+2), right? So why didnÕt you at \
least
do that
> tiny bit of the work before posting? There are two other
places where
> you should have seen common factors that are obvious and
easy to pull
> out. So why didnÕt you do those simple operations before
posting? Are
> you so terribly shy and afraid of making the slighest
mistake and being
> embarrassed that you would rather have somebody else do
your homework
> for you then even try to do it yourself?? Or are you just
plain too
> lazy to do your own work?
> Unfortunately Jeroen Boschma already posted a response with
most of
> that easy obvious stuff already done for you, and I presume
you looked
You are very right. I have a habit of posting a response to
questions I
think I can handle adequatly
(as a non-mathematician), but in this case I should not have
worked towards
the \final solution as
far as in my response. An equivalent example should be enough
to show the
ÔtrickÕ. Something to keep
in mind for the next time :)
> at and copied his work, so you have forever lost the chance
to \figure
> that sort of stuff out yourself. On your \final exam you
wonÕt have him
> to do your work for you, so I suggest you \find another
example from
> your book that looks about as dif\ficult as this example, and
try that
> example without any help, get as far along with the easy
stuff as you
> can, then post the question and your partial work here and
ask if
> thereÕs anything further you can do. ThatÕs \
the only way
youÕll learn
> how to do these sorts of factorizations and simpli\fications
of
> polymomial fractions.
===
Subject: Re: High school factorization of algebraic fraction
>> I was wondering how I could go about writing this
algebraic fraction
>> in the simplest factored form:
>> ((6a2+2a)/(4a+8))*((6a+12)/(a))
>> What steps should I take?
> When you say 6a2, do you mean 6 * a^2 ?
> IÕll assume the answer is yes.
get the little 2 for squared. I wasnÕt aware that it \
wouldnÕt
work.
> Well, before posting you should at least *try* to do the
obvious stuff.
> For example, do you see that 4a+8 has a common factor of 4?
So you can
> factor that as 4 * (a+2), right? So why didnÕt you at \
least
do that
> tiny bit of the work before posting? There are two other
places where
> you should have seen common factors that are obvious and
easy to pull
> out. So why didnÕt you do those simple operations before
posting? Are
> you so terribly shy and afraid of making the slighest
mistake and being
> embarrassed that you would rather have somebody else do
your homework
> for you then even try to do it yourself?? Or are you just
plain too
> lazy to do your own work?
I have maths revision sheets here that our teacher gave us to
use to study
for our upcoming mathematics exam. I tried this question for
about 4 hours
over the weekend. Every time I tried it, I came up with the
3(3a + 1). I
tried it several different ways, like expanding the top and
bottom lines and
attempting to factorize, etc. The reason that I kept trying
was that the
answer in the answer section on the sheets was:
(a(3a+1))/(3). I was
attempting to get this answer for a long time. I thought that
there must be
something harder about this question, because it was the last
question in
the exercise. I should have substituted values in for the
answer and the
question to make sure that it was actually a correct answer
in the answers
section. I decided that the people here might be able to
explain or whatever
how to get to the (a(3a+1))/(3). I wasnÕt aware of the
alt.math.undergrad
newsgroup that Stan Brown has told me, and I will be using
that group in
future. So yeah, sorry about wasting peoplestime and \
asking
questions that
I wasnÕt sure if the answer shown was correct anyway.
> Unfortunately Jeroen Boschma already posted a response with
most of
> that easy obvious stuff already done for you, and I presume
you looked
> at and copied his work, so you have forever lost the chance
to \figure
> that sort of stuff out yourself. On your \final exam you
wonÕt have him
> to do your work for you, so I suggest you \find another
example from
> your book that looks about as dif\ficult as this example, and
try that
> example without any help, get as far along with the easy
stuff as you
> can, then post the question and your partial work here and
ask if
> thereÕs anything further you can do. ThatÕs \
the only way
youÕll learn
> how to do these sorts of factorizations and simpli\fications
of
> polymomial fractions.
Well I thought that I had the grasp of the subject, and this
question came
out and stumped me when the given answer didnÕt match mine. \
I
have now gone
and tried all the questions in the algebraic exercise and I
am con\fident
that I know how to do them. Sorry that I didnÕt show my
partial working,
because I thought that it was wrong, but I guess now looking
back I should
have shown you my working so that you could point out where I
had made the
error if I had made an error. Sorry for wasting everyonesÕ
time...
===
Subject: Re: High school factorization of algebraic fraction
>I was wondering how I could go about writing this algebraic
fraction in the
>simplest factored form:
>((6a+2a)/(4a+8))*((6a+12)/(a))
The tops and bottoms can be multiplied:
[ (6a^2+2a) (6a+12) ] / [ (4a+8) a ]
Then just factor top and bottom normally, and look for common
factors
to remove.
Start with common monomials:
[ 2a(3a+1) * 6(a+2) ] / [ 4(a+2) a ]
You see that (a+2) is a factor of the whole top and the whole
bottom,
so itÕs gone:
[ 2a(3a+1) * 6 ] / [ 4a ]
And a is also a common factor, so itÕs gone:
[ 2(3a+1) * 6 ] / 4
Now on the top you have 2*6 = 12 = 4*3:
4*3(3a+1) / 4
and the common factor of 4 is divided out of top and bottom:
3(3a+1)
For really basic questions you might want to post to
alt.math.undergrad or even one of the k12 groups in future
--
Stan Brown, Oak Road Systems, Tompkins County, New York, USA
http://OakRoadSystems.com
A: Maybe because some people are too annoyed by top-posting.
Q: Why do I not get an answer to my question(s)?
A: Because it messes up the order in which people normally
read text.
Q: Why is top-posting such a bad thing?
===
Subject: Re: High school factorization of algebraic fraction
> Hi Group,
> I was wondering how I could go about writing this algebraic
fraction in
the
> simplest factored form:
> ((6a+2a)/(4a+8))*((6a+12)/(a))
> What steps should I take?
What steps have you taken so far?
===
Subject: Re: High school factorization of algebraic fraction
> Hi Group,
> I was wondering how I could go about writing this algebraic
fraction in
the
> simplest factored form:
> ((6a+2a)/(4a+8))*((6a+12)/(a))
> What steps should I take?
> James Midolo
You have:
(6a^2 + 2a)*(6a+12)
-------------------
(4a+8)*a
Try to move terms out of the ()-brackets, like:
(4a+8) = 4*(a+2)
(6a^2+2a) = 2a*(3a + 1)
YouÕll get:
2a*(3a + 1) * 6(a+2)
----------------------
4*(a+2) * a
Then eliminate equal terms up and below the division line,
recalling that
2*6/4=3 you get your
answer.
Jeroen
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===
Subject: FLTMA: FermatÕs Last Theorem and Modular Arithmetic
topic. I have no excuse. I will get to it. You can always use
Google Groups
with search terms dgoncz@ and Fermat for a rough scan.
I am using the AOL proportionally spaced newsreader to post
this list of
computer output. I will read this tonight in Google. How does
it look in
your
reader?
pa=phi(a) etc.
n
N a ta pa (c^n-b^n) mod a
b tb pb (c^n-a^n ) mod b
c tc pc (a^n+b^n) mod c
1 2 3 4 5 6 7 8 9 10 11
2 3 2 2 0 1
1 4 2 2 0 2
5 4 1 2
6 7 6 6 0 1 5 5 2 1
2 9 6 3 0 3 6
10 4 4 2 6 0 2
6 7 6 6 0 4 4 5 6 2
2 9 6 3 0 6 3
13 12 12 2 3 0 6 5 1 0 2 6 9 0 5
6 8 4 2 0 4
2 9 6 6 0 5 6 2 3 8
13 12 12 2 4 2 6 10 11 0 1 4 9 8 8
6 7 6 6 0 3 6 0 4 1
2 10 4 4 0 6 0 4
13 12 12 2 4 6 4 12 2 0 3 12 7 7 6
10 3 2 2 0 2
6 11 10 10 0 10 6 3 1 9 6 9 9 2
13 12 12 2 1 0 6 6 3 0 5 5 9 0 2
6 9 6 6 0 2 3 2 6 2
10 11 10 10 0 4 0 5 0 9 0 3 0 1
13 12 12 2 7 7 6 12 10 0 11 12 9 6 9
2 5 4 4 0 1 0 4
2 12 4 2 0 8
13 12 4 2 4 0 7
6 7 6 6 0 1 4 0 6 3
2 12 4 2 0 6
13 12 12 2 6 11 4 10 10 0 5 4 7 5 1
30 11 10 10 0 1 3 7 4 9 8 6 2 5
2 12 4 2 0 2
13 12 12 2 10 5 4 4 6 0 1 10 7 11 5
OK, thatÕs a little rough. The two smaller sequences should
be repeated to
extend to the alignment with the longer, and all three should
end with
zeroes.
(Zeros?) ItÕs a start.
The output from Mathcad is a matrix. It gets copied to the
Works
spreadsheet
where zeroes get blanked. That gets output as text and tabs
and converted
to
four space tabs with CLR Text to Spaces. That gets read into
the AOL text
editor in Courier font to check spacing and then pasted into
the chatty
proportional newsreader. Working with Outlook Express and
Outlook is a
little
frustrating, but I am on it.
I pasted a few zeroes back in after zeroes were blanked.
You see, it appears N is always even, intersecting with n
prime to give n =
2.
N is the location of the triple coincidence of 0, if it
occurs. What I do
is
generate vectors A, B, and C, mod a, b, and c, as above,
listing the dual
exponential congruential sequences to limits phi(a), phi(b),
and phi(c),
then
determine the actual periods ta, tb, and tc. I compute
lcm(ta,tb,tc). If
the
triple coincidence occurs, it must occur in this range. I
recompute A, B,
and C
in this new, larger range, and look for the coincidence
A.n+B.n+C.n=0, and
if
found, N=n, so then I add with MathcadÕs stack function to
the output
matrix.
I compute lcm(ta,tb,tc)/N to give Q, the number below N, and
forget to
include
that in the header. Sorry. I donÕt dare tinker with it now
that I am in the
proportional editor.
In tests to c=25, N ranged as high as 330.
Printed and faxed output of this list, larger lists, and the
Mathcad source
is
available for those interested in FLTMA.
I tolerance everything and tolerate everyone.
I love: Dona, Jeff, Kim, Kimmie, Mom, Neelix, Tasha, and Teri,
alphabetically.
I drive: A double-step Thunderbolt with 657% range.
I \fight terrorism by: Using less gasoline.
===
Subject: Re: SkolemÕs Paradox and why is math the way it is?
>>I mostly agree, but at least one of my physics professors
in college
>>considered treating the wave-function as real as wrong,
wrong, wrong.
>>It applies only to a statistical ensemble, not to just a
single
>>system. (!)
> DoesnÕt the Aspect experiment make that view untenable?
Quoting from [1]:
However, the low ef\ficiency of the detectors used in the
experiments
means that additional assumptions (essentially that those
photons
detected are a fair sample of the total ßux) have to be made
to test
the Bell inequality. If these assumptions are made, the
results are
found to rule out local realist theories, and to be in good
agreement
with the quantum predictions. Most physicists now accept that
quantum
theory is correct, and that local realism has to be abandoned.
However, other physicists, often known as the realists,
strongly
disagree. They question the additional [...]
Reference [1] presents an interesting biographical sketch of
John Bell.
[1] Andrew Whitaker:
John Bell and the most profound discovery of science,
Physics World, December 1998 URL:
David Bernier
===
Subject: Re: SkolemÕs Paradox and why is math the way it is?
> Axioms for FOL and ZFC are unambiguously stated at:
> <http://us.metamath.org/mpegif/mmset.html#axioms> (or just
using?) meta-logic, is this another name for SO logic?
No. The Metamath Proof Explorer merely follows the common
approach of
introducing a formal axiomatic \first-order theory by using
metavariables ranging over formulas and variables.
An example of this in the literature is given in MendelsonÕs
_ Introduction to Mathematical Logic _ (4th ed., p 69):
If B, C, and D are wfs of L, then the following are logical
axioms
of K:
(A1) B => (C => B)
(A2) (B => (C => D)) => ((B => C) => (B => D))
[...]
Compare to <http://us.metamath.org/mpegif/ax-1.html> and
<http://us.metamath.org/mpegif/ax-2.html>.
===
Subject: Re: SkolemÕs Paradox and why is math the way it is?
[...]
|> I think it will be mighty dif\ficult for you to order your
development
|> unless you admit the necessity of starting informally, for
at least a
|> brief time, or you decide to go formal all the way and not
care
about
|> such things as whether the theory has a model.
|
|Which theory are you talking about?
Any one that one intends to treat as foundational.
| Quine is discussing statements,
|and Hintikka has games, either of those seems \fine to treat
|informally, they are both just abstractions of VERBS, things
I can do
|personally.
You can perform uncomputable strategies and communicate them
to your allies, can you? I donÕt think so.
My sets are abstractions of ADJECTIVES. In order for an
adjective
to make sense, I donÕt necessarily have to be able to do
anything
besides understand what it means. IsnÕt that less \
problematic?
| And so it is disprovable and subject to experimental
|observation ultimately.
How?
Disproof requires a shared notion of what counts as proven
from
the premises in question. This ordinarily is done in an
informal
way, but if we want to resolve all disputes in a rigorous
way, we
ultimately want to formalize the theory in question. IF logic
hasnÕt been formalize as far as I know (and any \
formalization
would
necessarily be partial).
[...]
|> It seems, then, that we have found a dictionary between
whatever I
|> might have to say mathematically that concerns this large
submodel
|> of the cumulative hierarchy (the sets having rank less
than the
|> smallest inaccessible cardinal) can be translated into a
language
|> that you can understand!
|
|IÕm still not sure that there is such a submodel that \
\fits the
|standard interpretation.
Fine, lots of people doubt that too. The issue is whether the
properties that it would have, were it to exist, can be
discussed
in an unambiguous way. Since you believe in IF logic, you
agree
that they do; donÕt you have to?
| If I look at the set theory validities,
|i.e. the valid formulas N or T where T is an oFOL closed
formula and
|N is the IF-FOL translation of the SO negations of the SO
set theory
|axioms SO alternated together, then it can be true in all
models, but
|that doesnÕt mean that there is a model where N is actually
false. It
|just means that N or T is a validity.
I understand. The same could be said by a person who believes
in the
coherence of FOL as applied to models, but who doubts the
existence
of any in\finite sets.
|> |This is FINE for physics because in physics we ALSO play
veri\fication
|> |and falsi\fication games in the laboratory, so I can make
them match
|> But you donÕt play uncomputable strategies against the
universe,
|> so far as I know. IF-logic only works the way that it does
because
|> one is implicitly assuming the possibility of using
uncomputable
|> strategies.
|
|Where did computable come from?
As far as I know, you are unable to play uncomputable
strategies.
Second, IF logic develops a completely different semantics if
we
only consider computable strategies.
For example, suppose we play with rational numbers the game
corresponding to the sentence
(*) (A)(B) {[(EN) (x)(y) x>N -> y^2<>x^3+Ax+B] v
[(N) (Ex)(Ey) x>N & y^2=x^3+Ax+B].
This is just a sentence of \first-order logic. \
ItÕs regarded as
true
because the two subformulas in [] are negations of each
other, but
this is a nonconstructive fact. In order actually to be able
to win
every time, you would need an algorithm for determining
whether there
are arbitrarily large solutions in x and y to y^2=x^3+Ax+B.
I donÕt know offhand whether such an algorithm exists. I
believe it
to be a reasonably nontrivial question. Now take Joe Random
mathematician
and ask him whether (*) is true. Yes, of course, he says. Can
he always
play the game and win? He will inform you that believing in
(*) is not
based on a belief that we can *actually* play the game
associated with
it and win.
Some would say that (*) is true because a hypothetical
omniscient being
can play it and win every time, but thatÕs a completely
different story.
| The universe itself plays the part of
|the initial falsi\fier, the existance of the winning strategy
means you
|can defeat the universe at the game, every time.
If you can evaluate the winning strategy.
| If you can proof the
|validity, then you can show that you could win the game N or
T in
|any model. So IÕm not making the assumption that the proofs
construct
|all the validities, so why should I?
No, thatÕs beside the point.
| And you are totally forgeting
|that the game is usually a premise as well as a conclusion,
so to say
|that something is continuous, you might be more precise and
say that
|the game Ax Ay (~(0<y) or Ed Ar ~(|r|<d) or |f(x+r) - f(x)|
< |y|))
|is true for f, and the meaning of the sentance is that there
is a
|winning strategy for the game, you could THEN use that
strategy to
|prove other things about the function, like that Aa Ab (a<b &
|f(a)<f(b)) => (Az ~(a<z<b) or (Ec a<c<b & f(c)=z)), you
donÕt have to
|CONSIDER what the word computable means to do any of this.
But thatÕs not how you de\fined implication. The \
game
associated with
{premise}->{conclusion} can be won by the veri\fier only if he
can
\first determine either that he can win the game of {premise}
playing
the part of the falsi\fier, or that he can win the game of
{conclusion}
playing the part of the veri\fier. That makes it terribly hard
to play
the part of veri\fier successfully. The function in question
might have
a very small discontinuity in it hidden somewhere....
I said that it would be more consistent with what people
ordinarily
mean by -> to allow the veri\fier access to the strategy for
the premise,
but this is not part of IF logic.
|Validities is based partly on the fact that an oFOL
statement T always
|HAS a strategy for one side, so then you can assume a
strategy for one
|side and try to prove that a strategy exists so that the
E-team wins
|the game N or T and then you assume a strategy exists for
the other
|side and try to prove that a strategy exists for N or T for
the
|E-team. The word computable (whatever it means) never comes
up, the
|fact that T is oFOL and has a strategy for either side is
useful.
Now you are not learning how to play the game; you are
learning how
to deduce that a strategy would exist.
ThatÕs just a dodge. The fact is, these strategies you refer
to as
abstractly existing are highly idealized, and have relatively
little
to do with any strategies you can actually follow.
|Maybe you are bothered that there are two subjects, logic,
the study
|of formulas, and set theory the study of sentances T such
for a \fixed
|N (unkown to me) the formulas N or T are validities. Set
theory
|needs logic, it really does, I didnÕt think this was a
problem.
If you look in the mathematical reviews subject
categorization,
thereÕs a \field known as logic that is broad \
enough to include
set theory as a sub\field.
People do sometimes de\fine logic per se to be just the study \
of
logical validities. Your guess that this would bother me is
quite
a wild guess!
|> How else can it be true that either for every x, ~P(x), or
there
|> exists an x such that P(x)? The only way to win that game
is to
|> be able to search the whole domain of discourse to check
whether
|> any of the elements satis\fies P!
|
|I donÕt know what P(x) is, but saying Ax ~P(x) is true \
MEANS
that if
|you searched the universe that ~P(x) is true for every x, Ex
P(x) is
|true means that some choice of x makes P(x) true, whether
[Ax ~P(x)]
|or [Ey P(y)] is a validity or not is just a matter of
whether P(z) is
|true or false for every z, if it is, thatÕs enough, then \
you
know itÕs
|a validity. The point of a validity is that since it is true
for any
|play of the game in any model, it is subject to disproof,
thatÕs the
|best you can hope for. Proof in science is impossible, so
having a
|verb that serves as capable of disproof is the best you can
hope for.
|You canÕt perform every experiment in every place at every
time, but
|since the claims are universal, you can attempt disproof to
your
|heartÕs content.
I donÕt think this is correct. How do such sentences \
falsify?
Suppose
Dr. Smith has a theory that turns out to be equivalent to the
winnability
of some game played against the universe. But poor Dr. Smith
keeps losing
this game. Has his theory been falsi\fied? Maybe Smith is just
a poor
player of this game. If the only way he can win every time is
by being
able to solve problems without computable algorithms for
solving them,
I can hardly blame him for failing, can I?
I think integers are either prime or composite. If we play
the game
(n) {[(Es)(Et) 1<s,t<n & st=n] v [(s)(t) 1<s,t<n -> st<>n]}
on the natural numbers, the falsi\fier can probably ensure that
the
veri\fier will go to his grave before \finding the \
necessary s
and t
to win the game.
Keith Ramsay
===
Subject: Re: SkolemÕs Paradox and why is math the way it is?
[...]
|IÕve asked many times what the axioms are.
You know, I havenÕt seen it. I looked through a lot of the
messages
from you on this thread and didnÕt see any such question. I
have seen
lots of messages that suggest to me you think you already
know enough
about them to serve as a good critic.
One of your original questions:
Why do mathematicians like a nondescriptively complete axiom
system as
a basis?
You asked various times why we liked the axioms, hinting that
you
knew of some problem.
[...] how do we know everyone is doing the same math if the
axioms donÕt
describe real numbers uniquely?
Why all the blather about uncountability then?
It tends to be a bad sign when someone casually describes
standard
results in a \field as blather.
[Addressed to Torkel Franzen]
What axioms do YOU use?
He soon explained that ordinary mathematics is perfectly
adequate
for physics, and presumably thatÕs what he practices.
The axioms of ZF assert the existence of a set CALLED the
power set of
the naturals, but IÕve seen no proofthat this set has no
logical
correspondance to the naturals, in fact Skolem seems do give a
convincing case that it does and the the models of ZF donÕt
allow a
set to be created from this correspondance.
So you thought you understood the power set axiom well enough?
What I am saying is that the
\first order ZF set theory axioms donÕt prove the \
existence of
uncountably many subsets of the naturals.
I.e., J.E. knows his ZF well enough already to say what they
do
and do not prove.
IÕm not trying to do mathematics. IÕve said \
that all along.
What IÕm
doing is asking mathematicians about what the axioms THEY use
mean to
THEM so that I can decide if I was to use them in my PHYSICAL
models.
At this point it seemed pretty clear that what you were after
was
the MEANING of the axioms, i.e. a philosophical account, not
just a
statement of what they were.
There are only countably many of
them, but the others appear to be useless since for MANY MANY
years NO
ONE has proven theorems about them (since I havenÕt seen new
axioms
added to set theory to prove more subsets of the naturals
exist than
formerly could be proven to exist).
So J.E. has been keeping track of which axioms have been
added!
I HAVE written out de\finitions of countable models of ZF(C),
and they
are well-founded and pure, AND obviously incomplete in that
there are
things that should be sets that arenÕt in it.
J.E. has done model theory! Honestly, you claimed to have
written out
de\finitions of countable models of ZF(C), and now you tell us
you
never actually knew what the axioms were.
|> has been scarcely any strictly mathematical (as opposed to
|> philosophy of math) question in this discussion. You ask
things
|> like How do theorists know that the SEQUENCE to generate h
|> [PlanckÕs constant] exists in ZF? that \
doesnÕt have any
clear
|> meaning.
|
|Now you canÕt hold me responsible for whatever other people
bring to
|the discussion.
Of course. But for some reason their discussion of PlanckÕs
constant
prompted you to wonder whether it exists in ZF, a meaningless
phrase.
| I would wonder if ZF is the right theory in which to
|create a model that PREDICTS the value of h [PlanckÕs
constant],
Well thatÕs a heap of a lot better question.
The way to answer it is de\finitely not to worry about model
theory.
ItÕs to examine the kind of mathematics actually being used.
| but
|as far as IÕm considered the value we measure in the lab is
a rational
|number. The point is that people assert that everything is
in set
|theory, but they go OUTSIDE set theory to assert that, and I
donÕt
|know WHERE that brazen con\fidence comes from. ANYONE can just
ASSUME
|that all sets are in their theory.
There is no such concept as set in a theory. Completely
unde\fined!
You keep tossing it about as if it meant something, but it
doesnÕt.
The brazen con\fidence is the same chutzpah as leads us to
think that
when we say the bananas of the world, we mean, the bananas of
the
world, and not some subset of them.
Does it take a special argument to show that when we say
banana, we
mean banana? No. Is this a bizarre assertion? No.
[...]
|In physics if they had a wrong theory, theyÕd state at the
very very
|beginning that it was wrong, thatÕs very different than
saying that
|theorems are true for a whole semester.
They knew that we knew already. The professor made a blanket
statement
at the beginning of the course that every theory in physics
was an
approximation. I do not remember him reiterating this in
reference
to Newtonian mechanics.
|> There were a number of places where we had to fudge.
In\finities
|> appear in certain places that one just has to accept as
being
|> not quite right. The energy in the electric \field of a
charged
|> canÕt be correct; the product of the charge and the
electric
|> \field vector at it is unde\fined, because the \
electric \field
goes
|
|WHAT! Those can all be \fixed by being more careful.
There have been threads in sci.physics.research discussing
this--
itÕs not entirely trivial. I think calling it being careful \
is
to understate the case quite a bit. Professional physicists
have
wondered aloud whether classical E&M is consistent at all.
And 99% of your problems with mathematics could be cured by
being
half as careful as all that.
|NO ONE cares
|electrodynamics and I kept asking why donÕt we compute the
trajectory
You mean j?
| And the answer is that
|itÕs harder and no one cares.
This is assuming that your charges are small enough and
evenly enough
distributed that you can treat them like they form a
continuous current
density.
| But itÕs up to the people who care
|about something to make it work, the purpose of most physics
classes
|is not to teach you how to \find analytic solutions, \
itÕs to
develope
|physical intuition so that you can recognize correct results
and \fix
|problems caused by doing numerical approximations sloppily
and to get
|approximate answers so that you donÕt have to get the true
answer for
|every situation.
Aha! And what do you suppose your mathematics courses are
for, huh?
Oh, wait, sorry, weÕre supposed to be mindless drudges with
nothing
else to do but be as precise as possible. I keep forgetting
that.
[...]
|> What exactly is the Dirac delta function? Well, itÕs not
really
|> a function, and telling you what it precisely is, is
beyond the
|> scope of this course. :-) This willingness to work with an
ill-de\fined
|> concept is not just accepted; actual pride is taken by
physicists
|> in their willingness to dispense with precise de\finitions
and
|
|Distribution theory, if we want to have a pissing contest
about who
|has better teachers, then I can concede that many many bad
physics
|teachers exist, in fact thatÕs what IÕd like \
to change,
thatÕs my goal
|to improve physics teaching.
Terri\fic.
My point was not that these were bad teachers-- they were
good. My point
is that it was never a game of but you keep refusing to tell
me the
precise version of the story; IÕm going to whine until you
do, and if
I had taken that attitude, IÕd have about the same
relationship to
physics as you have to mathematics.
Keith Ramsay
===
Subject: Re: SkolemÕs Paradox and why is math the way it is?
|> | I consider
|> |math to be science.
|> Then try treating it like one.
|> The key objections to be made to a physical theory are
that its
|> predictions are observed to be incorrect, itÕs been
superceded by
|> a more comprehensive theory, and that itÕs needlessly
complex.
|> The corresponding objections can be made of a mathematical
theory
|> too, in principle, but I see you making an awful lot of
objections
|> that are of a completely different kind.
|
|In IF-FOL I can imagine an observational refutation of a
proposed
|validity (by demostrating game play inconsistent with that
|description), but I donÕt see how you can observe other
things to
|invalidate a theory, the theorems are treated formally if
you object
|to the English, and then someone will continue using the
English, so
|for most people itÕs just a waste of time to observe
anything in
|math.
Not at all. We can compute, for one thing. If a theory says a
computation will work out one way and it works out another way
(and we havenÕt made a mistake), then the theory must be
wrong.
|> Allow it to be just as sloppy and messy as physics is and
nearly
|> all of those remaining troubles are gone! If you are still
of the
|> impression that you canÕt tell what ZF is, try naming \
some
statement
|> which you either have trouble knowing how to formalize, or
donÕt know
|> whether itÕs an axiom, or a proof which you \
donÕt know
whether itÕs
|> valid, and we can surely clear it up for you.
|
|Is Ax ~Ef (Ea aex & {0,{0,a}}ef) & (An Ab (bex =>
({n,{n,b}}ef => (Ec
|cex & {nU{n},{nU{n},c}}ef & ceb)))) the foundation axiom? Is
it
|considered part of set theory? I have more questions like
this, but
|for all I know IÕm already kill-\filed by \
everyone.
No, that is not how the foundation axiom is usually written.
ItÕs
an odd formulation; where did you get it?
For one thing this representation of an ordered pair (a,b) as
{a,{a,b}}
is a bit unforunate, since the foundation axiom is needed to
prove that
if {a,{a,b}}={c,{c,d}} then a=c and b=d! I canÕt see offhand
whether
this creates any real troubles to have this *inside* the
foundation
axiom itself.
Also the role played by x seems a little obscure. For each f,
there
exists an x which contains the image of f (as a function),
i.e. the
set of b such that for some n, (n,b) is in f. So unless IÕm
missing
something, x is superßuous, and we can just say that there
does not
exist an f satisfying (Ea (0,a)ef) &
(An Ab ((n,b)ef => (Ec (n+1,c)ef & ceb)).
Foundation is usually expressed so as to say that each
nonempty set
has a member disjoint from it. I donÕt know what the \
original
form
it was written in. ItÕs always possible that \
itÕs been
rewritten
somewhat since it was \first stated as an axiom.
The way to go from the axiom of foundation to what youÕve \
got
is to
supose there were such an f, and then consider the set
S={f(n) : n
is a natural number}. Each member f(n) would share a member,
namely
f(n+1), with S. So by the usual axiom of foundation no such S
exists,
hence no such S.
[...]
|> We donÕt depend upon axiom systems in the way that you
imagine.
|> When someone claims that a result is a theorem, they mean
that it
|> has been proven, period. End of sentence. Not, proven
in... but
|> proven.
|
|What? You donÕt seriously mean proven without assuming any
standards
|of proof or axioms, do you?
I didnÕt say anything about standards of proof, but since \
you
ask, we
do of course have standards. They are just informal, not set
by a \fixed
axiom system.
I donÕt know how it is that people like you get so deeply
indoctrinated
into the idea that everything in mathematics depends upon a
choice of
axioms. Without getting a heavy dose of very consistently
worked out
formalist philosophy of mathematics along with it.
Keith Ramsay
===
Subject: Re: SkolemÕs Paradox and why is math the way it is?
> [...]
> |> We donÕt depend upon axiom systems in the way that you
imagine.
> |> When someone claims that a result is a theorem, they
mean that it
> |> has been proven, period. End of sentence. Not, proven
in... but
> |> proven.
> |What? You donÕt seriously mean proven without assuming \
any
standards
> |of proof or axioms, do you?
> I didnÕt say anything about standards of proof, but since
you ask, we
> do of course have standards. They are just informal, not
set by a \fixed
> axiom system.
But different axioms systems are different. For instance you
can have
a non-well founded set theory, then you usually have
collection
instead of repalcement, but in a normal set theory class a
proof of
collection is required.
> I donÕt know how it is that people like you get so deeply
indoctrinated
> into the idea that everything in mathematics depends upon a
choice of
> axioms. Without getting a heavy dose of very consistently
worked out
> formalist philosophy of mathematics along with it.
I took many classes where people said theorem and truth a lot
but
always turned out to really mean proof in ZFC. Personally, \
IÕm
actually surprsied to hear that most people donÕt turn out
like I did.
===
Subject: Re: SkolemÕs Paradox and why is math the way it is?
J.E.:
[...]
|IÕve never studied intuitionist logic, and I \
donÕt even know
what
|inconsistent is de\fined to be in intuitionist logic.
A set of statements S is inconsistent if for some statement
P, one can
intuitionistically deduce both P and ~P from the statements.
|Assuming y is
|in f(y) is true leads to ~f(y) is in f(y), which is a
|contradiction. Assuming ~y is in f(y) is true leads to
~~f(y) is
|in f(y), which is a contradiction. So we are lead to conlude
that y
|is in f(y) is neither true nor false. This is a problem if
you have
|an excluded middle, but otherwise, whatÕs the big deal. I
suspect
|from talking to you that intuitionist logic DOES have an
excluded
|middle, but that it has a limited power to discuss that fact
and a
|limited ability to infer from that fact.
Quite the opposite-- three-valued logic is essentially
classical
logic, but with a limited ability to use it. The distinctions
between a statement being true or not, and between it being
false or not are still essentially being treated as classical
distinctions, just with less of a chance to use them.
Three-valued logic is classical logic without negation as
applied
to pairs of propositions that contradict each other. Given
such a
pair (P, Q) where P->Q, we can decide to call it true when P
is
true, and false when Q is true, but indeterminate when both P
and Q are false. We de\fine the connectives like this:
(P1,Q1)&(P2,Q2) = (P1&P2,Q1vQ2),
(P1,Q1)v(P2,Q2) = (P1vP2,Q1&Q2),
~(P,Q)=(Q,P),
(P1,Q1)->(P2,Q2) = (Q1vP2, Q2&P1).
The vanilla quanti\fiers are similar to conjunction and
disjunction:
Ex (P(x), Q(x)) = (ExP(x), AxQ(x))
Ax (P(x), Q(x)) = (AxP(x), ExQ(x)).
The real effect of this is to put a restriction on the kind of
proposition that you can express in it. The propositions you
get
are essentially pairs of ones that can be de\fined without \
using
negation starting from the original propositions.
The weakening effect of this is not so serious if you have an
actual negation on atomic formulas, like Hintikka does. Each
statement in classical \first-order logic can be converted into
an equivalent one where the negations are all on atomic
sentences.
This is how he ends up having classical \first-order logic
embedded
in his logic.
ItÕs well-known that the usual proof of the \
unde\finability of
truth relies upon the language doing the de\fining being rich
enough to include a negation. The usual conclusion drawn from
the unde\finability of truth argument is that for each \
\fixed,
well-de\fined language there is another language that expresses
more than it does. I think thatÕs correct. But the richer
language
as usually de\fined applies negations to statements of the
original
language. ThereÕs a form of the diagonal argument for
predicates
on natural numbers-- if P_0(n), P_1(n),... is an enumeration
of
the predicates on natural numbers in some language, then there
is another one, Q(n) = ~P_n(n), that canÕt be expressed in \
the
language.
There are two ways to describe how Hintikka circumvents the
unde\finability of truth. One is to say that he realizes \
itÕs
appropriate to allow a generalization of \first-order logic
where
paradoxical sentences satisfying p<->~p can exist. The other
is
what IÕve been saying here, that the essential reason for \
his
success is that he rede\fines negation to be less expressive
than it usually is. The negation of this graph has at least
four connected components should be this graph has at most
three connected components as usual; itÕs just that Hintikka
ham-strings negation to keep us from expressing propositions
like that. This second explanation seems more illuminating to
me than HintikkaÕs.
Intuitionist logic represents a more fundamental departure
from
classical logic than three-valued logic does. Instead of just
pasting together two distinctions, each of which looks just
like
an ordinary classical proposition, to get a trichotomy,
intuitionist logic permits one to make an array of more subtle
distinctions. Classical logic has the connective & and ~ and
the quanti\fier for all that are pretty close to intuitionist
ones. However, given those three, the remaining connectives
and
quanti\fiers of classical logic are redundant:
P v Q can be de\fined as ~(~P&~Q).
P -> Q can be de\fined as ~(P&~Q).
(Ex) P(x) can be de\fined as ~(Ax)~P(x).
In intuitionist logic, however, none of these three is
redundant.
They all are needed to express additional shades of meaning
that
classical logic and 3-valued logic wash out. The Ex and v
are taken in a constructive sense. Ex means that one can
effectively
get an x. AvB is treated similarly, and means that one can
effectively get a choice of side, left or right of the
disjunction
that holds true. Intuitionist implication is more subtle
still.
In classical logic, starting from n propositions P1,...,Pn,
we get
2^n possible assignments of truth-values. For any
propositional
expression involving the P1,...,Pn, we get an assignment of a
truth
value to each of these assignments (i.e., the truth-table for
the
expression). That gives us 2^(2^n) possibilities. Every
formula of
classical propositional calculus in n variables is equivalent
to one of those 2^(2^n).
In three-valued logic, this argument no longer applies
because one
doesnÕt have excluded middle. But instead one simply has \
three
truth-values. The situation is similar, except that not all of
the 3^(3^n) possibilities for assignments of truth-values are
allowed.
Intuitionist logic, however, canÕt be exhausted by any \
\finite
set of
possible truth-values to assign to propositions. ItÕs a \
famous
metatheorem about intuitionist logic there given a
proposition p,
there is an in\finite sequence of propositions that can be
expressed
in terms of p, none of them necessarily equivalent to each
other.
ItÕs true that there isnÕt a special \
distinction between ~P
is true
and P is not true or P is false, because in intuitionist logic
there wasnÕt a need to rede\fine ~P to mean \
something unusual.
So
the usual proof of the unde\finability of truth still works, \
and
~(P and ~P) is valid, as is ~(~P and ~~P). But neither of them
entails P or ~P, which means that we can in principle
determine
the truth or falsity of P, which we canÕt.
|> I think youÕre avoiding this conclusion by considering it
possible
|> for some statement to be true if and only if it is not
true. I
|> understand how IF-logic permits such a thing to occur, but
itÕs
|> not convincing to me that this makes good sense.
|
|And I donÕt understand what IT MEANS to not have an exluded
middle and
|disallow that.
The law of excluded middle says that p or ~p holds for every
p.
The absence of a p for which p is true if and only if it is
false
means that ~(p<->~p) holds for every p. A system having the
one rule
and a system having the other rule are not the same thing. I
just
happen to know of a very well-known system where the second
rule
holds but not the \first.
|We are coming from different worlds and I donÕt know
|the basis for your ideas, while you know that my meanings
are derived
|from the semantics of games. So itÕs a bit unfair for me to
be
|explaining things to you.
I donÕt agree.
ItÕs rare for a person to be familiar with IF logic, and \
since
weÕve started the discussion, IÕve learned \
more about it than
I
knew to begin with. This is more than youÕre entitled to
expect.
I havenÕt asked you to explain things that I already know.
If you want to make claims about how the law of excluded
middle
affects the nature of a logical system, I think itÕs \
entirely
reasonable of me to mention facts about the most well-known
and
most heavily studied logical system lacking the law of
excluded
middle.
YouÕve done an unusual amount of complaining about people
failing
to explain things to you. I can appreciate your complaining
if you
have had professors who were paid to explain things to you,
but were
stingy with their time in of\fice hours, or something like
that, but
nobody here is being paid to do anything, and in fact weÕve
used a
bunch of our own spare time to try to explain things.
As far as leaving the basis for my ideas obscure, I think
youÕre
expecting a different kind of explanation from me that is
reasonable
to expect.
IÕm more willing than most people are to approach issues
either from
a formalist or a realist perspective. To put it crudely,
there is a
kind of chicken-and-egg problem: either you start with
sentences or
with objects. The formalist treats as fundamental the
sentences, but
in doing so he is stuck having to treat them as uninterpreted
sentences,
because he lacks any real objects for them to refer to. The
realist
coming before any formalization, he has to discuss them to
begin with
in informal terms.
The realist approach makes people uneasy, because they donÕt
feel like
there has been enough of an explanation of the nature of the
objects
referred to by the fundamental, unde\fined terms. In our case,
these
would be the sets. I suspect your perception that I havenÕt
explained
to you where my ideas come from is largely due to your
discomfort with
the unde\fined term set. You appear to be craving a kind
explanation
of what set means that would no longer treat it as a
fundamental
term. But if one wants to treat oneÕs mathematical \
statements
as having
a de\finite meaning, one needs some terms whose meanings are
only
informally explained; itÕs just how it is.
I think itÕs no better to regard IF logic as a (realistic)
foundation.
The way most people understand the meanings of the terms
involves a
belief in the existence of these domains on which the games
are played,
as well as the strategies of the players which might or might
not be
winning strategies. Domains are essentially classes, and the
strategies
are essentially functions on them. The strategies are not
necessarily
actually playable in practice (they often are uncomputable),
so weÕre
not gaining extra support from some understanding of our
actual
game-playing abilities.
ItÕs possible to be consistently formalistic, but it means
that a lot
of the questions you ask no longer make any sense. ItÕs just
incoherent
to ask whether your model has all the sets it should have, as
a
formalist, because you are not basing your theory on the idea
that
there exists a model of it. If you believe in the existence
of a model
of your theory, then you are being somewhat of a realist. I
donÕt
think itÕs very sensible to be partially a realist but to
brush the
fact under the rug by pretending that all of your terms are
de\fined
inside of one or the other theory. If your notion of model is
de\fined inside a theory, then use that theory, because \
itÕs
more
fundamental than the theory assumed to have a model.
|And there is going to be huge problems
|because we de\fine implication differently, I use a stronger
version
|than use, it is more expressive and says more, but therefore
is has
|fewer rules of inference.
Your implication is stronger and satis\fies fewer rules of
inference,
but it is not more expressive. Just as in classical logic,
itÕs
redundant; one could use ~AvB instead. Intuitionist
implication is
not redundant, and allows us to express the existence of a
connection
between the premise and conclusion. A classical implication
is supposedly
true always by virtue of some property holding of one of the
sides;
either A is false, which makes the implication true
regardless of what
B is, or B is true, which makes the implication true
regardless of what
A is. Intuitionist implication can hold without either side
necessarily
making it true by itself.
| So my biconditional says A <-> ~A means that
|(~A or ~A) & (A or A) which is logically equivalent to ~A &
A which
|means that A cannot be true or false, full stop.
Why not, in addition to this implication, include the real
one?
Allow me to say that, given a winning strategy for A, I can
give
you a winning strategy for B.
|> Once we have a logic with three truth-values, true, false
and
|> indeterminate, I donÕt see how it can be invalid for me \
to
start
|> talking about which of the three bins a sentence falls
into. The
|> claim that a certain sentence simply fails to be true,
i.e., is
|> either false or indeterminate, appears to make logical
sense. ItÕs
|> just not a claim that can be expressed in IF-logic.
|
|Huh? Truth is about a winning stratgey in all models, same
with
|falsity.
I donÕt think this is standard terminology. Certainly in
classical
\first-order logic, one de\fines the notion of the \
sentence being
true
in a model. It can be true in some models and not in others.
IÕm saying that the sentences of IF logic intentionally
exclude such
model-relative claims as in this model, the veri\fier has no
winning
strategy for the game associated with S. ThatÕs what the
negation
of S should mean. The language is weakened by our not being
allowed
to say such a thing.
| Being neither can mean totally different things. It could
|mean that it has a winning strategy for one team in one
model one for
|the other team in another model, or it could mean that there
is a
|model where the sentance has no winning strategy for either
side.
YouÕre providing good examples here of the kind of thing \
that
one
wants to say when discussing IF logic, that doesnÕt really \
\fit
inside of it. The statement in this model, such-and-such
sentence
has no winning strategy for either side isnÕt expressible in
IF
logic. Everything we say about winning strategies is part of
the
metalanguage, since winning strategy doesnÕt have a
translation
into the logic itself.
This is why I donÕt see it as a fundamental theory; almost
anyone who
believes it is a coherent theory also believes in the
existence of
things like winning strategies that donÕt belong inside it.
| Why
|does it make sense to lump these cases together?
I donÕt.
| What we care about
|is truth, which means winning in all models.
I thought winning was everything?
But seriously, suppose I claim to care about whether a
sentence
lacks a winning strategy for either player in all models.
WhatÕs
wrong with that, aside from not belonging to the formalism?
| For instance if N is the
|negation of an axiom and T is a theorem of the axioms, then
N or T
|is a validity, true, true in all models, in all worlds. And
there are
|TWO ways to negate that CONCEPT, to say false in all worlds
or to
|say not the case that it is true in all worlds. IF-logic
does the
|FIRST, because the point is that you write SENTANCES and
then you
|assert that their truth means something about THE DOMAIN OF
DISCOURSE,
|this is what we do in physics, we state the world is such
that T is
|true, itÕs what philosophers do. To discuss anything else
invovles
|actually quantizying over possible worlds, which I didnÕt
think people
|were still serious about doing.
But youÕve been quantifying over all possible models
throughout the
discussion, here. How does it even make sense for you to
divide up
the possible cases the way you just did, if itÕs
inappropriate to
quantify over possible domains of discourse.
Quantifying over possible worlds has a different connotation
from
quantifying over all possible models. The former is an
extension of
modal logic; the latter is set theory.
|> This is really what I want Hintikka to tell me: why am I
mistaken
|> when I think IÕve made an assertion such as the domain of
discourse
|> is \finite that is true *exactly* when a certain sentence of
IF-logic
|> fails to be true. In what way is this an incoherent
sentence? It
|> seems as though he simply ham-strings his theory to make
it impossible
|> to say certain things in it.
|
|ItÕs an funny difference of opinions because you think he
ham-stringed
|his theory, and it seems to me that you want him to
ham-string his
|theory when he hasnÕt.
Yes, he has. IÕve given more than one example. \
ItÕs not just
that
there are things that canÕt be expressed in the language,
since thatÕs
always true. ItÕs that the things that canÕt \
be expressed are
so much
like the ones that can be, aside from being prevented. Even
just to
allow games where to win a player needs to make a \finite
number of
moves of a certain kind, without a \fixed upper limit, extends
the
language enourmously at essentially no cost.
|Validities are what we care about, things true
|in all worlds,
Not everybody. Many of us care about other things as well. If
you
want to keep saying what we care about is... you should say
why we shouldnÕt care about other things as well.
[...]
|(~A or B) is stronger than A=>B
|~A is stronger than it is not the case that A is true, so we
can
|say things that you canÕt otherwise say in FOL.
Preventing yourself from saying it is not the case that A is
true is
not an advantage. The only reason ~A fails to be expressible
is that
itÕs been reduced to the status of a second proposition that
happens
to contradict A. There can be different AÕs, where there
exists a winning
strategy in the same domains, but whose negations are
different, just
because.
The reason you can say things in IF logic that canÕt be said
in ordinary
FOL is the use of these independence requirements on
variables. You could
just as well include the ordinary ~ and => of FOL, and it
would give you
a stronger theory, equivalent to second-order logic.
|ItÕs is a WEAKER claim about the universe of discourse, it
merely says
|consider the worlds where the theorems are true not consider
the
|worlds where the (second order) axioms are true. You want
nonsesnse?
| It IS nonsense to say consider a FO universe of discourse
where the
|SO axioms are true if you just want the FO axioms and oFOL
theorems,
|then IF-FOL is useless.
Which second-order axioms? Why not consider those domains on
which a
collection of second-order axioms are true? You havenÕt
presented
anything like a cogent argument against the good sense of
this. The
natural numbers N={0,1,2,...} are de\finable up to isomorphism
as the
model of a set of second order axioms. WhereÕs the big
problem with
this? Why should I limit myself to some less expressive
language?
|If you want to translate SO axioms into FO
|language without assuming a universe of discourse that
INCLUDES SO
|entities.
A strategy for one of these games is already a second-order
entity,
since it is made up of functions on the domain of discourse.
How are
you really hoping to get away from that?
| Then instead of considering the universe where all the
|axioms are true you should instead consider the universe of
discourse
|where all the theorems are true, do you really not get it?
I donÕt think youÕre explaining your point \
very well here.
You write
as if you think you have a really strong argument, but I
donÕt see
that you do.
| The IF
|logic claim is more powerful because when you actually
translate SO
|axioms into IF-FOL you get statements that do NOT have
contradictory
|negations in second-order logic, that is because they simply
do NOT
|have winning strategies for either side, so INSTEAD of
considering
|universes where you can VERIFY the truth of the axioms, you
consider
|the ones where it is impossible to VERIFY the negation of
the axioms.
IÕm not sure of which translation you have in mind.
Overall, itÕs especially unclear what the actual \
bene\fit is
supposed
to be. You accept the coherence of IF logic, which is \fine,
but seems
to require accepting the meaningfulness of terms like
strategy and
model or domain. Having accepted those, the reason for having
special qualms about second-order logic is obscure.
|It is simply astonding that the ordinary proofs of theorems
carry over
|to IF-FOL validities because it is a much much stronger
claim to say
|that the theorems are all true in models where the axioms
are not
|false, then to MERELY say they are true when the axioms are
true,
|because the INTENDED axioms are NOT true in any model.
ThatÕs what YOU think, but do you have any argument for it?
|IÕm trying to
|explain why what you are asking for is nonsense, but I \
donÕt
know if
|you can see it.
The problem isnÕt that I canÕt see it; the \
problem is that it
isnÕt nonsense.
|> Wittgenstein gave an example of an incoherent description:
when itÕs
|> \five oÕclock on the Sun. He imagines someone \
asking, what
does that
|> mean? and the answer is, Just like what it means to be \five
oÕclock
|> here-- but on the Sun. I can imagine that some of the
things I think
|> and talk about are confused in sort of this way: it only
seems to me
|> that IÕm considering a well-de\fined \
proposition. The only
way I can
|> see how it would make sense to consider a system like
IF-logic to be
|> ultimate is if I were confused in such a manner about the
things that
|> I say that appear not to be expressible in IF-logic.
|
|The claims you want to make (contradictory negations of
IF-FOL
|formulas) are actually either IF-FOL formulas (in the
special case
|where the original formula was actually logically equivalent
to an
|oFOL formula) or the negations are ACTUALLY SO statements
and they
|require a CHANGE of the universe of discourse to INCLUDE SO
entities.
|If you want to stay FO, then IÕd be hard pressed to come up
with some
|stronger and more expressive logic than IF-FOL, but maybe
there is
|one. I think Hintikka had an extended IF-logic.
I think IF-logic is \first order in only a phony way. Its
semantics
appears to require having an intuitive understanding of what a
strategy is, which is already a second-order notion. The fact
that
one does not allow direct references to strategies (although
they
clearly do exist) doesnÕt strike me as an advantage of
IF-logic.
Keith Ramsay
===
Subject: Re: SkolemÕs Paradox and why is math the way it is?
|I should also point out that the Aspect experimentÕs
|results were published in December of 1982, and I donÕt
remember
|for sure whether the remark was made before or after the
|paper was published. So maybe he changed his mind
|afterward.
Now that I think about it, the Aspect results must have been
published within a year or two before this professor made
his remark.
Keith Ramsay
===
Subject: Re: SkolemÕs Paradox and why is math the way it is?
Sorry about the lag; I think IÕve been trying to do too many
things
all together. IÕll probably have to taper off my \
contribution
soon.
|> Most mathematics, incidentally, uses only a relatively
uncontroversial
|> portion of set theory. People deal with things like sets
of real
|> numbers all the time, but not so often the parts that
depend on
|> (say) the axiom of replacement.
|
|Since we havenÕt proven that ALL the set theory axioms \
taken
together
|are consistent, then youÕd expect that if a smaller subset
works for
|physics, that someone would have tried to prove that that
smaller set
|of axioms was consistent. Is there such a proof?
There is indeed a proof that the axioms of set theory, without
replacement, are consistent. However! The proof uses the axiom
of replacement. By GoedelÕs second incompleteness theorem, a
system
like Z (the system without the axiom of replacement) canÕt
prove
its own consistency, so some additional axiom is needed, of
course.
The most familiar proof works by showing the existence of a
model
of Z. The most obvious model of Z is V_{omega+omega}, which
is the
\first part of the cumulative hierarchy: Let V_0 be the empty
set,
and by induction let V_{n+1} be the set of all subsets of V_n
for
n=0,1,2,.... These are all \finite sets, which can be written \
as
\finite strings if need be. TheyÕre known as the \
hereditarily
\finite
sets, since not only are they \finite, but their members are
\finite,
the members of their members are \finite, and so on.
Then let V_{omega} be the set of all hereditarily \finite sets.
This
is a countably in\finite set. It requires the axiom of \
in\finity
to
prove that it exists, but its elements can be put into
correspondence
with the natural numbers. Incidentally V_{omega} is a model
of all the
axioms of ZFC except for the axiom of in\finity.
De\fine V_{omega+(n+1)} to be the set of all subsets of
V_{omega+n}
inductively. Then \finally let V_{omega+omega} be the union of
all
of the V_{omega+n} for natural numbers n. That last step is
the only
step in the proof where the axiom of replacement is needed.
[...]
|> I donÕt think there is a be-all theory.
|
|IsnÕt being a be all theory part and parcel of the standard
|interpretation that everything that could exist for anyone
that is
|small enough to be a set, is a actually a set?
No. It seems as though youÕre confusing an interpretation
with a
theory. One can believe that one has a coherent
interpretation in
which when one says set, it really means any kind of set. That
doesnÕt mean that one has a be all theory of what \
theyÕre
like.
ItÕs like the difference between believing that you know \
what
an
apple is, and knowing everything that there is to know about
apples.
|The only reason to
|insist on this rather than just that enough sets exist to
satisfy the
|axioms is because one wants to pretend like one can have an
|everything, where the universe of discourse of standard
|interpretation set theory is superior to all other universe
of
|discourses.
No. There are larger domains of discourse. Set theory is
merely a
relatively large one.
|Otherwise why is that interpretation consider necissary or
even
|standard?
It seems to me that youÕre putting the cart before the horse
again.
I consider the primary purpose of the axioms to be to
investigate a
domain of inquiry systematically, *not* to de\fine the limits
of the
domain of inquiry.
Unless you have a good reason to think that itÕs impossible
to talk
about all real numbers (and I donÕt think you do), \
itÕs a very
strange suggestion that we should talk only about some of
them.
|> |But IF
|> |logic avoids having in\finite regresses into higher-order
logics so
|> |that we CAN sit down and discuss how you make theories,
so isnÕt that
|> |worth considering?
|> What in\finite regress into higher-order logic is there for
anyone?
|
|Like you say in your previous post, you need SO set theory
to de\fine
|the strongly inaccessible cardinal, in order to get a
faithful model
|of set theory, but once you introduce SO set theory, people
will want
|the other sets too, because the whole POINT of introducing
that
|cardinal was to get all the sets that were missing in
previous
|models. You arenÕt succeeding at getting all the sets.
I donÕt think I said you needed second-order set theory to
de\fine a
strongly inaccessible cardinal. It only needs a \first-order
de\finition
in terms of the epsilon relation. As usual, of course, to say
that
an element of a model satis\fies this de\finition as \
relativized
to the
model doesnÕt mean the same thing as saying that its a
cardinal
satisfying the de\finition.
I think you need to distinguish between various senses of
get, here,
pertaining to the scope of variables, the language as a
whole, and the
axioms.
If I say that all real numbers are either <0, >0 or =0, then
my
statement implicitly contains a quanti\fier for a variable
ranging
over all the real numbers. The statement succeeds in getting
all
the reals in the sense that it quanti\fies over them.
If I say that I can de\fine any arbitrary algebraic number,
then IÕve
gotten them all in second sense. This sense is relative to my
language,
since what de\finitions I can provide depend on how rich my
language is.
If all I have are the elementary school operations of +,-,*,/
and maybe
simple roots x^(1/n), then my language is too weak to de\fine
all of them.
There is no language (with \finite expressions) strong enough
to get
all the real numbers in this sense.
If I say that I can prove the existence of a weak
inaccessible cardinal,
then I have gotten it in a third sense, which depends on what
can be
proven (which depends on which axioms are accepted).
These three senses are in order of increasing narrowness. In
order to
prove that something exists, I need to be able to describe
it. In order
to describe it, I need to have (implicitly at least)
variables that
range over a domain that includes it.
When you say get, you often seem to be sliding between these
senses.
You seem often to be trying to treat them as if they were the
same
thing. You donÕt seem to see any problem with treating the
narrowest
sense (what we can prove to exist) as if it should somehow be
the same
as the range of our quanti\fiers. I donÕt see any \
point in
doing that.
If we have been talking about real numbers, I donÕt see any
point in
deciding to assume that we are *always* talking about some
subset of
de\finable or provably existing real numbers instead. The only
reason I
can think of for wanting to do that, not just sometimes but
generally,
would be if there was somehow a serious problem with the
concept of
real number, some genuine ambiguity or incoherence.
[...]
|> You canÕt say that a graph has three connected \
components,
in it.
|
|Do you have a citation for that result, or better yet can
you state
|your de\finition of graph and connected component?
I donÕt have a citation for it offhand. \
ThereÕs nothing
special
about three, by the way. I used that because I was thinking I
could
go on to point out that This graph has at least three
connected
components as well as this graph has at least four connected
components were both expressible in IF logic, but not this
graph
has (exactly) three connected components.
Connectedness is a familiar example of a non-\first-order
property of
a structure. ItÕs not expressible in IF logic because IF \
logic
extends \first-order logic by permitting an existential
quanti\fier
over subsets of the structure (in effect).
Those are the sigma-1-1 properties. But connectedness is a
pi-1-1
property, which is how it escapes being \first-order \
de\finable.
ItÕs
usually de\fined as meaning that any two vertices are joined by
a path.
ThatÕs equivalent to the nonexistence of a way to divide the
graph
into two nonempty disjoint subsets, with no edges connecting
any
vertex in the one with any vertex in the other.
Lemme see if I can sketch a proof that itÕs not also
sigma-1-1.
Suppose there is a game (associated to a sentence in IF-logic)
with a winning strategy for the veri\fier on an \
in\finite
connected
graph where the degree of the vertices is bounded above by
some
natural number n. For simplicity, we can take a set of
vertices
indexed by the integers (including negative integers) where
the
edges connect adjacent vertices. The winning strategy
consists of
a collection of functions f(a1,...,a_m) for different values
of m.
I claim that there exist disconnected graphs where the \
veri\fier
also has a winning strategy.
First, a simple example. IÕm pretty sure that the graph
consisting
of two disjoint copies of the original graph is an example.
Take
the points (x,y) in the plane where y=0,1 and x is an
integer, and
join the points (x,y) and (x+1,y) by edges to form the graph.
IÕm
suffering a little writerÕs block on the proof, though.
Second, pulling out the big guns. The original graph has just
two
relations on it, xEy meaning that x and y are joined by an
edge,
and x=y. Now augment the structure by adding the functions f
that
correspond to the veri\fierÕs winning strategy. \
ThatÕs now a
model
of the \first-order sentence saying that the \
veri\fier wins the
game
regardless of what the falsi\fier plays.
The upward Lowenheim-Skolem theorem says (as a special case)
that if
a \first-order sentence has an in\finite model, it \
also has an
uncountable
model. So the sentence saying the functions f are a winning
strategy
for the veri\fier, and that all of the vertices have degree 2,
also holds
true for some functions fon a graph with uncountably many
vertices.
Since a connected components of a graph whose vertices have
degree 2
is always countable, this graph with uncountably many
vertices is
disconnected.
The same proof works just as well for the sentence, this
graph has
three connected components. Any game associated with a
sentence of
IF logic that can be won on a graph with three components can
always
be won on a graph with more components.
To me this just reveals something missing in IF logic. We can
understand
nearly as easily what it means to be able to win the
following game:
the refuter picks two vertices, and to win the veri\fier has to
present
vertices one at a time, each connected to the previous one,
and starting
with the \first vertex given by the refuter get to the other
one. ItÕs true
that this involves the notion of a \finite sequence of moves,
but I donÕt
see how that can be much worse than the kind of arbitrary
strategy
allowed the players in a game associated with an IF logic
sentence.
IF logic just is so limited that we canÕt say it. We can \
even
say what I
would call the REAL negation of the claim that the graph is
connected:
that it can be divided into two nonempty parts that arenÕt
connected to
each other.
[...]
|> |Every model of set theory lacks a set that should exist
as much as the
|> |alleged uncounted real should exist.
|> If by model you mean a set with an epsilon relation on it,
then
|> this is correct, but people often mean by model either a
set *or*
|> a proper class with an epsilon relation on it. The
cumulative
|> hierarchy does not lack a set that should exist-- it
consists
|> by de\finition in all the well-founded pure sets.
|
|This is really hard to discuss non-circularly. The words
structure,
|class, function, set, collection, relation all have
de\finitions in
|the theory and to use the same words outside of the theory
is begging
|for confusion.
I think trying to force them to be theory-dependent in an
inconsistent
way is begging for deeper confusion.
I donÕt think there is such a thing as a different \
de\finition
of
function, for example, in the theory. ItÕs possible that you
are
alluding to relativizing some of these concepts to models,
but you
need to distinguish theories from models.
|What do you want to take as given?
IÕm pretty ßexible about what to take as given, so long \
as
weÕre
consistent with it.
We can start the formal development by taking set as an
unde\fined
term, either believing that we know a de\finite meaning for it,
or by
merely proceeding as though it does. To remain consistent
with such
a starting point, however, makes a lot of statements
nonsensical, like
saying that this domain of sets lacks a set that should exist.
If by model, you mean a set having an epsilon relation and so
on, then
itÕs circular to try to de\fine set relative to \
model, because
this
sense of model depends on the concept of set already. Set
needs
to have a meaning that doesnÕt depend on models. We do not
need to start
out with a model in this sense of some theory-- itÕs \
circular
to try
to start that way. ItÕs true of this kind of model that \
there
are sets
that are not members of it; it does not contain itself, for
instance, and
it is by de\finition a set.
If by model you mean something broader, like what
philosophers sometimes
call a domain of discourse, then we can, if you like, call
the domain of
sets that we start out with a model, but then there is no
longer any
sense in saying that our starting model is missing any sets.
We have just
de\fined the domain of discourse to consist of all such objects
that we
are going to be calling sets.
When you asked questions about deciding to use the minimal
model,
since the minimal model is de\fined in terms of the concept of
set,
whatever you meant by minimal model was dependent on some
prior
concept of set. Certainly if you want to use such a model for
physics
there isnÕt an inconsistency, but you are still stuck with
the fact that
you started out with one brand of set theory, and then
created a second
kind that depends on the original kind. Most people \figure
that they
are better off just sticking to whatever kind of set they
started out
with.
| We could have a
|third person in the game, and have the third person start
talking,
|saying a in M, aea in A, ain M, \
aÕeain A, aeain E,
aÕea in A,
|aÕin M, \
aÕÕeaÕin A, \
aÕÕeain A, \
aÕÕea in A, \
aÕeaÕin
A, aeaÕin
|A, aÕÕin M, \
aÕÕÕeaÕ\[CapitalO\
Tilde]in A, \
aÕÕÕea in A, \
aÕÕÕeaÕ\[CapitalO\
Tilde] in A,
aÕÕÕeaÕ\[Capital\
OTilde] in
|A, aeaÕÕin A, \
aÕeaÕÕin A, \
aÕÕeaÕÕ\[CapitalO\
Tilde] in A, ... and but
where the
|third person chooses freely whether to say xey in A or xey
in E,
These little scenarios where you describe some outside source
generating
a structure incrementally strike me as having so little about
them that
is analogous to the way mathematics actually works or how we
talk about
it, that I can hardly think of anything to say about them.
[...]
|I donÕt understand your claim about the cumulative
hierarchy, once you
|\finish the model, someone can take the standard
interpretation and
|say that some sets are missing,
Why?
|isnÕt V=L considered restrictive by
|mathematicians?
The best guess I can come up with here is that youÕre
confusing two
de\finitions of hierarchies here, the de\finition of \
the
cumulative
hierarchy (whose members constitute V) and the de\finition of
the
constructive hierarchy (whose members constitute L).
Keith Ramsay
===
Subject: Re: SkolemÕs Paradox and why is math the way it is?
> Sorry about the lag; I think IÕve been trying to do too
many things
> all together. IÕll probably have to taper off my
contribution soon.
> |> Most mathematics, incidentally, uses only a relatively
uncontroversial
> |> portion of set theory. People deal with things like sets
of real
> |> numbers all the time, but not so often the parts that
depend on
> |> (say) the axiom of replacement.
> |Since we havenÕt proven that ALL the set theory axioms
taken together
> |are consistent, then youÕd expect that if a smaller \
subset
works for
> |physics, that someone would have tried to prove that that
smaller set
> |of axioms was consistent. Is there such a proof?
> There is indeed a proof that the axioms of set theory,
without
> replacement, are consistent. However! The proof uses the
axiom
> of replacement. By GoedelÕs second incompleteness theorem,
a system
> like Z (the system without the axiom of replacement) canÕt
prove
> its own consistency, so some additional axiom is needed, of
course.
> The most familiar proof works by showing the existence of a
model
> of Z. The most obvious model of Z is V_{omega+omega}, which
is the
> \first part of the cumulative hierarchy: Let V_0 be the empty
set,
> and by induction let V_{n+1} be the set of all subsets of
V_n for
> n=0,1,2,.... These are all \finite sets, which can be written
as
> \finite strings if need be. TheyÕre known as \
the hereditarily
\finite
> sets, since not only are they \finite, but their members are
\finite,
> the members of their members are \finite, and so on.
> Then let V_{omega} be the set of all hereditarily \finite
sets. This
> is a countably in\finite set. It requires the axiom of
in\finity to
> prove that it exists, but its elements can be put into
correspondence
> with the natural numbers. Incidentally V_{omega} is a model
of all the
> axioms of ZFC except for the axiom of in\finity.
> De\fine V_{omega+(n+1)} to be the set of all subsets of
V_{omega+n}
> inductively. Then \finally let V_{omega+omega} be the union
of all
> of the V_{omega+n} for natural numbers n. That last step is
the only
> step in the proof where the axiom of replacement is needed.
> [...]
> |> I donÕt think there is a be-all theory.
> |IsnÕt being a be all theory part and parcel of the \
standard
> |interpretation that everything that could exist for anyone
that is
> |small enough to be a set, is a actually a set?
> No. It seems as though youÕre confusing an interpretation
with a
> theory. One can believe that one has a coherent
interpretation in
> which when one says set, it really means any kind of set.
That
> doesnÕt mean that one has a be all theory of what \
theyÕre
like.
> ItÕs like the difference between believing that you know
what an
> apple is, and knowing everything that there is to know
about apples.
> |The only reason to
> |insist on this rather than just that enough sets exist to
satisfy the
> |axioms is because one wants to pretend like one can have an
> |everything, where the universe of discourse of standard
> |interpretation set theory is superior to all other
universe of
> |discourses.
> No. There are larger domains of discourse. Set theory is
merely a
> relatively large one.
> |Otherwise why is that interpretation consider necissary or
even
> |standard?
> It seems to me that youÕre putting the cart before the
horse again.
> I consider the primary purpose of the axioms to be to
investigate a
> domain of inquiry systematically, *not* to de\fine the limits
of the
> domain of inquiry.
> Unless you have a good reason to think that itÕs \
impossible
to talk
> about all real numbers (and I donÕt think you do), \
itÕs a
very
> strange suggestion that we should talk only about some of
them.
> |> |But IF
> |> |logic avoids having in\finite regresses into higher-order
logics so
> |> |that we CAN sit down and discuss how you make theories,
so isnÕt that
> |> |worth considering?
> |> |> What in\finite regress into higher-order logic is there
for anyone?
> |Like you say in your previous post, you need SO set theory
to de\fine
> |the strongly inaccessible cardinal, in order to get a
faithful model
> |of set theory, but once you introduce SO set theory,
people will want
> |the other sets too, because the whole POINT of introducing
that
> |cardinal was to get all the sets that were missing in
previous
> |models. You arenÕt succeeding at getting all the sets.
> I donÕt think I said you needed second-order set theory to
de\fine a
> strongly inaccessible cardinal. It only needs a \first-order
de\finition
> in terms of the epsilon relation. As usual, of course, to
say that
> an element of a model satis\fies this de\finition \
as
relativized to the
> model doesnÕt mean the same thing as saying that its a
cardinal
> satisfying the de\finition.
> I think you need to distinguish between various senses of
get, here,
> pertaining to the scope of variables, the language as a
whole, and the
> axioms.
> If I say that all real numbers are either <0, >0 or =0,
then my
> statement implicitly contains a quanti\fier for a variable
ranging
> over all the real numbers. The statement succeeds in
getting all
> the reals in the sense that it quanti\fies over them.
> If I say that I can de\fine any arbitrary algebraic number,
then IÕve
> gotten them all in second sense. This sense is relative to
my
language,
> since what de\finitions I can provide depend on how rich my
language is.
> If all I have are the elementary school operations of
+,-,*,/ and maybe
> simple roots x^(1/n), then my language is too weak to de\fine
all of them.
> There is no language (with \finite expressions) strong enough
to get
> all the real numbers in this sense.
> If I say that I can prove the existence of a weak
inaccessible cardinal,
> then I have gotten it in a third sense, which depends on
what can be
> proven (which depends on which axioms are accepted).
> These three senses are in order of increasing narrowness.
In order to
> prove that something exists, I need to be able to describe
it. In order
> to describe it, I need to have (implicitly at least)
variables that
> range over a domain that includes it.
> When you say get, you often seem to be sliding between
these senses.
> You seem often to be trying to treat them as if they were
the same
> thing. You donÕt seem to see any problem with treating the
narrowest
> sense (what we can prove to exist) as if it should somehow
be the
same
> as the range of our quanti\fiers. I donÕt see \
any point in
doing that.
> If we have been talking about real numbers, I donÕt see \
any
point in
> deciding to assume that we are *always* talking about some
subset of
> de\finable or provably existing real numbers instead. The
only reason I
> can think of for wanting to do that, not just sometimes but
generally,
> would be if there was somehow a serious problem with the
concept of
> real number, some genuine ambiguity or incoherence.
> [...]
> |> You canÕt say that a graph has three connected
components, in it.
> |Do you have a citation for that result, or better yet can
you state
> |your de\finition of graph and connected component?
> I donÕt have a citation for it offhand. \
ThereÕs nothing
special
> about three, by the way. I used that because I was thinking
I could
> go on to point out that This graph has at least three
connected
> components as well as this graph has at least four connected
> components were both expressible in IF logic, but not this
graph
> has (exactly) three connected components.
> Connectedness is a familiar example of a non-\first-order
property of
> a structure. ItÕs not expressible in IF logic because IF
logic
> extends \first-order logic by permitting an existential
quanti\fier
> over subsets of the structure (in effect).
> Those are the sigma-1-1 properties. But connectedness is a
pi-1-1
> property, which is how it escapes being \first-order
de\finable. ItÕs
> usually de\fined as meaning that any two vertices are joined
by a path.
> ThatÕs equivalent to the nonexistence of a way to divide
the graph
> into two nonempty disjoint subsets, with no edges
connecting any
> vertex in the one with any vertex in the other.
> Lemme see if I can sketch a proof that itÕs not also
sigma-1-1.
> Suppose there is a game (associated to a sentence in
IF-logic)
> with a winning strategy for the veri\fier on an \
in\finite
connected
> graph where the degree of the vertices is bounded above by
some
> natural number n. For simplicity, we can take a set of
vertices
> indexed by the integers (including negative integers) where
the
> edges connect adjacent vertices. The winning strategy
consists of
> a collection of functions f(a1,...,a_m) for different
values of m.
> I claim that there exist disconnected graphs where the
veri\fier
> also has a winning strategy.
> First, a simple example. IÕm pretty sure that the graph
consisting
> of two disjoint copies of the original graph is an example.
Take
> the points (x,y) in the plane where y=0,1 and x is an
integer, and
> join the points (x,y) and (x+1,y) by edges to form the
graph. IÕm
> suffering a little writerÕs block on the proof, though.
> Second, pulling out the big guns. The original graph has
just two
> relations on it, xEy meaning that x and y are joined by an
edge,
> and x=y. Now augment the structure by adding the functions
f that
> correspond to the veri\fierÕs winning strategy. \
ThatÕs now a
model
> of the \first-order sentence saying that the \
veri\fier wins the
game
> regardless of what the falsi\fier plays.
> The upward Lowenheim-Skolem theorem says (as a special
case) that if
> a \first-order sentence has an in\finite model, it \
also has an
uncountable
> model. So the sentence saying the functions f are a winning
strategy
> for the veri\fier, and that all of the vertices have degree
2, also holds
> true for some functions fon a graph with uncountably \
many
vertices.
> Since a connected components of a graph whose vertices have
degree 2
> is always countable, this graph with uncountably many
vertices is
> disconnected.
> The same proof works just as well for the sentence, this
graph has
> three connected components. Any game associated with a
sentence of
> IF logic that can be won on a graph with three components
can always
> be won on a graph with more components.
> To me this just reveals something missing in IF logic. We
can understand
> nearly as easily what it means to be able to win the
following game:
> the refuter picks two vertices, and to win the veri\fier has
to present
> vertices one at a time, each connected to the previous one,
and starting
> with the \first vertex given by the refuter get to the other
one. ItÕs
true
> that this involves the notion of a \finite sequence of moves,
but I donÕt
> see how that can be much worse than the kind of arbitrary
strategy
> allowed the players in a game associated with an IF logic
sentence.
> IF logic just is so limited that we canÕt say it. We can
even say what I
> would call the REAL negation of the claim that the graph is
connected:
> that it can be divided into two nonempty parts that arenÕt
connected to
> each other.
> [...]
> |> |Every model of set theory lacks a set that should exist
as much as
the
> |> |alleged uncounted real should exist.
> |> |> If by model you mean a set with an epsilon relation
on it, then
> |> this is correct, but people often mean by model either a
set *or*
> |> a proper class with an epsilon relation on it. The
cumulative
> |> hierarchy does not lack a set that should exist-- it
consists
> |> by de\finition in all the well-founded pure sets.
> |This is really hard to discuss non-circularly. The words
structure,
> |class, function, set, collection, relation all have
de\finitions in
> |the theory and to use the same words outside of the theory
is begging
> |for confusion.
> I think trying to force them to be theory-dependent in an
inconsistent
> way is begging for deeper confusion.
> I donÕt think there is such a thing as a different
de\finition of
> function, for example, in the theory. ItÕs possible that
you are
> alluding to relativizing some of these concepts to models,
but you
> need to distinguish theories from models.
> |What do you want to take as given?
> IÕm pretty ßexible about what to take as given, so long \
as
weÕre
> consistent with it.
> We can start the formal development by taking set as an
unde\fined
> term, either believing that we know a de\finite meaning for
it, or by
> merely proceeding as though it does. To remain consistent
with such
> a starting point, however, makes a lot of statements
nonsensical, like
> saying that this domain of sets lacks a set that should
exist.
> If by model, you mean a set having an epsilon relation and
so on, then
> itÕs circular to try to de\fine set relative to \
model, because
this
> sense of model depends on the concept of set already. Set
needs
> to have a meaning that doesnÕt depend on models. We do not
need to start
> out with a model in this sense of some theory-- itÕs
circular to try
> to start that way. ItÕs true of this kind of model that
there are sets
> that are not members of it; it does not contain itself, for
instance, and
> it is by de\finition a set.
> If by model you mean something broader, like what
philosophers sometimes
> call a domain of discourse, then we can, if you like, call
the domain of
> sets that we start out with a model, but then there is no
longer any
> sense in saying that our starting model is missing any
sets. We have just
> de\fined the domain of discourse to consist of all such
objects that we
> are going to be calling sets.
> When you asked questions about deciding to use the minimal
model,
> since the minimal model is de\fined in terms of the concept
of set,
> whatever you meant by minimal model was dependent on some
prior
> concept of set. Certainly if you want to use such a model
for physics
> there isnÕt an inconsistency, but you are still stuck with
the fact that
> you started out with one brand of set theory, and then
created a second
> kind that depends on the original kind. Most people \figure
that they
> are better off just sticking to whatever kind of set they
started out
> with.
> | We could have a
> |third person in the game, and have the third person start
talking,
> |saying a in M, aea in A, ain M, \
aÕeain A, aeain E,
aÕea in A,
> |aÕin M, \
aÕÕeaÕin A, \
aÕÕeain A, \
aÕÕea in A, \
aÕeaÕin
A, aeaÕin
> |A, aÕÕin M, \
aÕÕÕeaÕ\[CapitalO\
Tilde]in A, \
aÕÕÕea in A, \
aÕÕÕeaÕ\[CapitalO\
Tilde] in A,
aÕÕÕeaÕ\[Capital\
OTilde] in
> |A, aeaÕÕin A, \
aÕeaÕÕin A, \
aÕÕeaÕÕ\[CapitalO\
Tilde] in A, ... and but
where the
> |third person chooses freely whether to say xey in A or xey
in E,
> These little scenarios where you describe some outside
source generating
> a structure incrementally strike me as having so little
about them that
> is analogous to the way mathematics actually works or how
we talk about
> it, that I can hardly think of anything to say about them.
> [...]
> |I donÕt understand your claim about the cumulative
hierarchy, once you
> |\finish the model, someone can take the standard
interpretation and
> |say that some sets are missing,
> Why?
> |isnÕt V=L considered restrictive by
> |mathematicians?
> The best guess I can come up with here is that youÕre
confusing two
> de\finitions of hierarchies here, the de\finition \
of the
cumulative
> hierarchy (whose members constitute V) and the de\finition of
the
> constructive hierarchy (whose members constitute L).
> Keith Ramsay
I want to compliment you on your recent posts, I think they
have been
really
excellent.
IÕve learned some things from them, mostly interfaces to
speci\fic terms and
words that then offer ready venues into relatively more well
developed
mathematical topics than mine. IÕm talking about ready
interfaces for my
theory, of course.
I have a question about the universe of discourse, could you
please expand
what you mean when you say that there are larger realms of
discourse than
set
theory?
Yeah, V=L.
Consider your cumulative hierarchy in the theory with
ubiquitous ordinals,
V{n+1} = P(V{n}) = succ(V{n}) = n+1. Particularly, consider
them with
their description as Z, the set of integers.
Surely, that gets into N-1, etcetera, omega - 1, because \
itÕs
talk about
negative one. The idea is to \find the mechanistic operation
upon the set
representing the positive and the set representing the
negative integer to
sum them. I think they might be from fully decorating the
ordinals,
instead
of using the naked ordinals, the more and less ornate
ordinals in the
ubiquitous ordinals. As well, in the theory of ubiquitous
naturals, then I
promote splitting in\finity in half. Half of the integers are
positive.
About the graph problem there, just embed them in matroids.
I must consider Replacement vis-a-vis the con\firmator, and \
UÕs
mask.
Consider Russell.
Skolem: why is math the way it is?
Ross F.
===
Subject: anti-cantorian probability theory and dart throwing
Suppose the reals are countable. :)
Then R = Q / Irr, where Irr is RQ.
If somebody throws a dart at a dartboard,
what is the probability that the x-coordinate
of the point where the dart lands is in Q (the rationals),
and why?
David Bernier
===
Subject: Re: anti-cantorian probability theory and dart
throwing
> Suppose the reals are countable. :)
> Then R = Q / Irr, where Irr is RQ.
> If somebody throws a dart at a dartboard,
> what is the probability that the x-coordinate
> of the point where the dart lands is in Q (the rationals),
> and why?
> David Bernier
I think the probability depends on the velocity and distance
of the
dart. On a massless spaceship moving at an acceleration of 1g
and no
air, then the dart would move in a parabola, so if you threw
it with a
rational velocity in the right direction from the right
distand, then
it would hit a rational point. In other circumstance it would
be
irrational. So the question is begged back towards you
providing a PD
of the phase space of the center of mass of the dart, and
from that,
we can kinematically transform the PD of the dartÕs position
at one
time into a PD on the x coordinate of the dart later, and
from that
answer your probability question.
===
Subject: Re: anti-cantorian probability theory and dart
throwing
> Suppose the reals are countable. :)
Starting from a false premise, anything that follows is
worthless. If
you want anyone to read the rest of your message, I suggest
you not
start off with a blatantly false premise. I suggest you avoid
all
mention of the reals. I suggest you devise some model that
doesnÕt
involve real numbers.
===
Subject: Re: anti-cantorian probability theory and dart
throwing
Suppose the reals are countable. :)
> Then R = Q / Irr, where Irr is RQ.
So Irr = empty set , which means that pie (=3.14159...), an
irrational
number, does not exist. This implies that the area of your
darts
board, which equals (pie*r^2) (where r is the radius) does
not exist
either. The probability that you would hit your dartsboard
would
therefore be virtually zero, and so the answer to your
question is
straigtforward: Only intelligent questions can have a
meaningfull
answer.
thomas
*-----------------------*
www.GroupSrv.com
*-----------------------*
===
Subject: Re: anti-cantorian probability theory and dart
throwing
> Suppose the reals are countable. :)
>> Then R = Q / Irr, where Irr is RQ.
> So Irr = empty set ,
the mere fact that R is a countable set containing Q it does
not follow
that RQ is empty, but from the fact that R is both countable
and
uncountable you can conclude that Bertrand Russell is the
Pope.
--
Dave Seaman
Judge YohnÕs mistakes revealed in Mumia Abu-Jamal ruling.
<http://www.commoncouragepress.com/index.cfm?action=book&
bookid=228>
===
Subject: Re: anti-cantorian probability theory and dart
throwing
> Suppose the reals are countable. :)
>>Then R = Q / Irr, where Irr is RQ.
> So Irr = empty set , which means that pie (=3.14159...), an
irrational
> number, does not exist. This implies that the area of your
darts
> board, which equals (pie*r^2) (where r is the radius) does
not exist
> either. The probability that you would hit your dartsboard
would
> therefore be virtually zero, and so the answer to your
question is
> straigtforward: Only intelligent questions can have a
meaningfull
> answer.
> thomas
IÕm not saying that the reals are countable. (hence the :) \
).
For those in this newsgroup (a vocal few) who donÕt accept \
the
uncountability of the real numbers, and declare that
the reals are countable, all I can say is
that it would be interesting to see how they would
answer the question...
I accept the validity of CantorÕs proof that R is \
uncountable.
David Bernier
===
Subject: Re: anti-cantorian probability theory and dart
throwing
posting-account=Glvc4AwAAADzVCZ73XnxpzMhXir6xVzs
The probability is 0. The number of irrationals in any given
interval
is uncountable, while the number of rationals is countable.
The probability is 1. A dart has \finite width and thus its
point will
span in\finitely many rationals no matter where it lands.
Take your pick.
===
Subject: Re: anti-cantorian probability theory and dart
throwing
>Suppose the reals are countable. :)
>Then R = Q / Irr, where Irr is RQ.
>If somebody throws a dart at a dartboard,
>what is the probability that the x-coordinate
>of the point where the dart lands is in Q (the rationals),
>and why?
This probability is 17. (Simple proof by contradiction
snipped.)
>David Bernier
************************
David C. Ullrich
===
Subject: Higher degree congruences
itÕs easy to \find congruences like
x = a (mod N)
and there could also be solutions for quadratic congruences
like
x^2 = a (mod N)
but is there something to solve congruences like
x^2 + ax = b (mod N)
or x^a + x^b + cx = d (mod N)
where a, b, c, d are integers
That is, what about degrees higher than 2 congruences?
Only a curiosity.
===
Subject: Re: Higher degree congruences
> itÕs easy to \find congruences like
> x = a (mod N)
> and there could also be solutions for quadratic congruences
like
> x^2 = a (mod N)
> but is there something to solve congruences like
> x^2 + ax = b (mod N)
> or x^a + x^b + cx = d (mod N)
> where a, b, c, d are integers
> That is, what about degrees higher than 2 congruences?
> Only a curiosity.
Well, for x^2 + ax = b (mod N)
\first, factorise N into its prime factors.
Then we have to solve x^2 + ax = b (mod p)
for every prime p dividing N.
for every p^t dividing N. Now use the Chinese remainder
theorem to piece together the solution for N.
To solve x^2 + ax = b (mod p)
simply complete the square to reduce it to y^2 = A(mod p).
I donÕt think there is an algorithm to solve congruences of
higher degree
than 2.
For example, how does one solve x^3+ax+b= 0(mod p), other
than by trial?
Ray Steiner
===
Subject: Re: Higher degree congruences
> x^2 + ax = b (mod N)
Well, you can allways use normal procedure of solving
quadratic equation
- make a quadrat. It will give You solution of particular
equation, but
wonÕt rather tell You much in general...
sirix.
===
Subject: Re: Higher degree congruences
>> x^2 + ax = b (mod N)
> Well, you can allways use normal procedure of solving
quadratic equation -
> make a quadrat.
Not allways the normal procedure...
for instance how do you manage to make a quadrat with
x^2 + 7x + 2 = 0 (mod 8)
This is easy, but not with the usual make a quadrat...
Because 2 has no inverse modulo 8, you canÕt write as
(x + 7/2)^2 +...
You need a different method.
--
philippe
(chephip at free dot fr)
===
Subject: Re: Confused about DFT and Fourier Series and
Fourier Transform?
Check http://www.ee.vt.edu/~ee4624ss/week4.pdf
> I am confused by the four transforms in Signal & Systems...
> The Continuous Time Fourier Transform(CTFT) is most
understandable; DFT
and
> Fourier Series alone are individually recoginizable and
understandable...
> Not sure about how does DTFT kick in...
> Anyway, remembering all of these four transformsformulas
are already
very
> headache... very easily got confuse one with another...
> Even worse, homework and test problems often asks for
conversion among
these
> four transforms...
> Given a signalÕs CTFT, how do you get DFT for N-point? How
does the DFT
> compare to the Fourier Series(looks to me they are all
discrete spectrum,
> etc.) so on and so forth, how are they related and how to
get one from
> another?
> Are there any good resources that clearly demonstrate the
relationship and
> conversion among these 4 transforms?
===
Subject: Two Interesting Second Order Linear Recurrence
Generators
Two interesting second order linear recurrence sequences are
{ 2, 2, 0, 1 } and
{ 2, 4, 0, 7 }.
Can you tell why?
They come from FLT, that is why. Each sequence is made of the
values of the
dual exponential generator most strongly associated with
a^n + b^n = c^n. It is:
(a^n + b^n) mod c. There are of course, two other generators.
The speci\fic values are
a=3; b=4; c=5 and
a=5; b=12; c=13.
Can you solve for a.0 and a.1 in the following equations?
x.2 = a.0 * x.0 + a.1 * x.1 (mod c)
x.i = a.0 * x.(i-2) + a.1 * x.(i-1) (mod c)
where {x} is one of the set of values above and c is the
associated 5 or
13?
For such generators a.0 and a.1 are, as you might expect, mod
c, that is,
they
can have any value from 0 to c-1.
I have tried this and found multiple solutions. Should there
be none, one,
or
more than one for each Pythagorean triple (a,b,c)?
Note there are two terms, these are second order generators,
two is prime
and a
Galois \field of order two exists. Also, two and the trivial
one are the
only
possible values for n in FLT. And I have two feet. :)
Coincidence? Perhaps...
You decide.
Can you solve this?
Basically itÕs
2,2 * A = 0
2,0 * A = 1
0,1 * A = 2
1,2 * A = 2
and you solve for A, a 2x2 matrix of con\figuration:
0 a.0
1 a.1
At some point we start leaving out the mod c.
I tolerance everything and tolerate everyone.
I love: Dona, Jeff, Kim, Kimmie, Mom, Neelix, Tasha, and Teri,
alphabetically.
I drive: A double-step Thunderbolt with 657% range.
I \fight terrorism by: Using less gasoline.
===
Subject: Re: Two Interesting Second Order Linear Recurrence
Generators
That is,
0 == 2 * a.0 + 2 * a.1 (mod 5)
1 == 2 * a.0 + 0 * a.1 (mod 5)
and with that zero in the second equation we have
2 * a.0 mod c == 1
LetÕs try a.0 = 0. Nope. 1. Nope. 2. Nope. 3. Yep. Any more?
4. Nope. 5.
Nope.
6. Nope.
a.0 = 3
2*3 + 2*a.1 == 0 mod 5
2 looks good. Any more? 0. Nope. 1. Nope. 2. Well, we have 2.
3. Nope. 4.
Nope.
5. Nope. 6. Nope.
a.1 = 2; a.0 = 3.
>{ 2, 2, 0, 1 }
2 == 3*0 + 2*1 mod 5.
2 == 3*1 + 2*2 mod 5.
It works!
IÕve shown that for *a* dual exponential congurential
sequence associated
with
*a* solution to FLT there is *exactly one* linear recurrence
generator and
associated matrix. And there are mentions of Galois \fields in
the literature
on
these sequences.
Can you show that for *any* dual exponential congurential
generator or
sequence there is an associated unique linear recurrence
generator?
I tolerance everything and tolerate everyone.
I love: Dona, Jeff, Kim, Kimmie, Mom, Neelix, Tasha, and Teri,
alphabetically.
I drive: A double-step Thunderbolt with 657% range.
I \fight terrorism by: Using less gasoline.
===
Subject: Re: Two Interesting Second Order Linear Recurrence
Generators
>a.1 = 2; a.0 = 3.
Mathcad veri\fies this is the right solution, and shows that
for the dual
subtractive generator, there is more than one equivalent
recurrence
generator
with identical sequence.
I tolerance everything and tolerate everyone.
I love: Dona, Jeff, Kim, Kimmie, Mom, Neelix, Tasha, and Teri,
alphabetically.
I drive: A double-step Thunderbolt with 657% range.
I \fight terrorism by: Using less gasoline.
===
Subject: Re: logic of the Cantorian followers mind
> : <snipped> :> The
> :> mainstream accepts the existence of objects that canÕt
be \finitely
> :> described.
> : Exactly what objects are those?
> Most of the real numbers. Most of the sets of integers.
> Most of the languages over any alphabet.
> Stephen
ThatÕs a \finite description of those things. A \
poor one, but
\finite.
===
Subject: Re: logic of the Cantorian followers mind
Originator: joshp@xoxy.net (joshp)
>> :> The mainstream accepts the existence of objects that
canÕt be
\finitely
>> :> described.
>> : Exactly what objects are those?
>> Most of the real numbers. Most of the sets of integers.
>> Most of the languages over any alphabet.
>ThatÕs a \finite description of those things.
No, those are \finite descriptions of the *class* of each of
those
things. But by CantorÕs argument, those classes each contain
elements
that are not \finitely describable.
--
Josh Purinton
===
Subject: Re: logic of the Cantorian followers mind
> :> The mainstream accepts the existence of objects that
canÕt be
> \finitely
> :> described.
> : Exactly what objects are those?
> Most of the real numbers. Most of the sets of integers.
> Most of the languages over any alphabet.
>>ThatÕs a \finite description of those things.
> No, those are \finite descriptions of the *class* of each of
those
> things. But by CantorÕs argument, those classes each
contain elements
> that are not \finitely describable.
> --
> Josh Purinton
Which real numbers canÕt be \finitely described?
===
Subject: Re: logic of the Cantorian followers mind
Originator: joshp@xoxy.net (joshp)
> Which real numbers canÕt be \finitely \
described?
Let Ident be a set of identi\fiers (\finite strings \
over a
countable
alphabet), and let X be a class with uncountably many
elements, such as
the class of all reals. Let F be a function from Ident into X.
Then there are uncountably many elements of X that are not in
the
range of F.
--
Josh Purinton
===
Subject: Re: logic of the Cantorian followers mind
>> Which real numbers canÕt be \finitely \
described?
> Let Ident be a set of identi\fiers (\finite \
strings over a
countable
> alphabet), and let X be a class with uncountably many
elements, such as
> the class of all reals. Let F be a function from Ident into
X.
> Then there are uncountably many elements of X that are not
in the
> range of F.
> --
> Josh Purinton
Does CantorÕs diagonal proof show this? What does \
identi\fiers
have to do with CantorÕs diagonal proof?
===
Subject: Re: logic of the Cantorian followers mind
Originator: joshp@xoxy.net (joshp)
>> Let Ident be a set of identi\fiers (\finite \
strings over a
countable
>> alphabet), and let X be a class with uncountably many
elements, such as
>> the class of all reals. Let F be a function from Ident
into X.
>> Then there are uncountably many elements of X that are not
in the
>> range of F.
> Does CantorÕs diagonal proof show this?
Given a function F from a countable set into an uncountable
set X,
diagonalization demonstrates the existence of an element of X
that is
not in the range of F. The absence of uncountably many
elements of X
from the range of F is an easy corollary.
> What does identi\fiers have to do with CantorÕs \
diagonal
proof?
Ident is a countable set.
--
Josh Purinton
===
Subject: Re: logic of the Cantorian followers mind
> Let Ident be a set of identi\fiers (\finite \
strings over a
countable
> alphabet), and let X be a class with uncountably many
elements, such as
> the class of all reals. Let F be a function from Ident into
X.
> Then there are uncountably many elements of X that are not
in the
> range of F.
>> Does CantorÕs diagonal proof show this?
> Given a function F from a countable set into an uncountable
set X,
> diagonalization demonstrates the existence of an element of
X that is
> not in the range of F. The absence of uncountably many
elements of X
> from the range of F is an easy corollary.
>> What does identi\fiers have to do with \
CantorÕs diagonal
proof?
> Ident is a countable set.
Why countable? Why does the alphabet need to be countable?
===
Subject: Re: logic of the Cantorian followers mind
> What does identi\fiers have to do with CantorÕs \
diagonal
proof?
>> Ident is a countable set.
> Why countable? Why does the alphabet need to be countable?
Are you talking about the alphabet, or about the set of
strings that can
be formed from that alphabet? The English alphabet has only
26 letters,
but thereÕs a countable in\finity of strings of \
letters.
And no, using an in\finite alphabet wonÕt change \
anything, as
long as itÕs
a countable in\finity.
If you are going to talk about uncountable alphabets, then
you may as
well let each real number be a character in this alphabet.
According
to that scheme, each real number has a description that is one
character long, but the character is no more describable than
the
number is. What have you gained?
--
Dave Seaman
Judge YohnÕs mistakes revealed in Mumia Abu-Jamal ruling.
<http://www.commoncouragepress.com/index.cfm?action=book&
bookid=228>
===
Subject: Re: logic of the Cantorian followers mind
Originator: joshp@xoxy.net (joshp)
> Why does the alphabet need to be countable?
Are you familiar with the Church-Turing thesis?
--
Josh Purinton
===
Subject: Re: logic of the Cantorian followers mind
>> Why does the alphabet need to be countable?
> Are you familiar with the Church-Turing thesis?
> --
> Josh Purinton
IÕve been told CantorÕs proof has nothing to \
do with
computability. I donÕt \find it strange that I \
was able
to lead you here though.
===
Subject: Re: logic of the Cantorian followers mind
:> :> The mainstream accepts the existence of objects that
canÕt be
:> \finitely
:> :> described.
:> : Exactly what objects are those?
:> Most of the real numbers. Most of the sets of integers.
:> Most of the languages over any alphabet.
:>>ThatÕs a \finite description of those things.
:> No, those are \finite descriptions of the *class* of each of
those
:> things. But by CantorÕs argument, those classes each
contain elements
:> that are not \finitely describable.
:> --
:> Josh Purinton
: Which real numbers canÕt be \finitely \
described?
Most of them.
Stephen
===
Subject: Re: logic of the Cantorian followers mind
<co0h2q$poj$2@msunews.cl.msu.edu>
<41Qod.89279$T02.75468@twister.rdc-kc.rr.com>
<q2Qod.452553$D%.116021@attbi_s51>
<ymQod.89282$T02.74358@twister.rdc-kc.rr.com>
<co0kpi$t2i$1@msunews.cl.msu.edu>
Discussion, linux)
> : Which real numbers canÕt be \finitely \
described?
> Most of them.
Name one.
--
Jesse F. Hughes
C is for Cookie. ThatÕs good enough for me.
Cookie Monster
===
Subject: Re: logic of the Cantorian followers mind
>> : Which real numbers canÕt be \finitely \
described?
>> Most of them.
>Name one.
I donÕt see any smiley or other indication of irony there, \
so
I assume
youÕre considering that to be a reasonable request. But in
fact itÕs not a
reasonable request to be asking for a description of
something that by
de\finition lacks a description, nÕest-ce pas?
The answer most of them is thoroughly accurate: the ones that
do have
\finite descriptions are a vanishingly small fraction of the
total.
BTW, this fact is used in ShanonÕs elegent \
information-theory
proof that
most codings are completely random (and hence perfect); itÕs
only (and
exactly) the ones that we can actually generate that arenÕt!
--
---------------------------
| BBB b Barbara at LivingHistory stop co stop uk
| B B aa rrr b |
| BBB a a r bbb | Quidquid latine dictum sit,
| B B a a r b b | altum viditur.
| BBB aa a r bbb |
-----------------------------
===
Subject: Re: logic of the Cantorian followers mind
<co0h2q$poj$2@msunews.cl.msu.edu>
<41Qod.89279$T02.75468@twister.rdc-kc.rr.com>
<q2Qod.452553$D%.116021@attbi_s51>
<ymQod.89282$T02.74358@twister.rdc-kc.rr.com>
<co0kpi$t2i$1@msunews.cl.msu.edu>
<87act7fyz1.fsf@phiwumbda.org>
<co325r$b8v$1@lust.ihug.co.nz>
Discussion, linux)
> : Which real numbers canÕt be \finitely \
described?
> Most of them.
>>Name one.
> I donÕt see any smiley or other indication of irony there,
so I
> assume youÕre considering that to be a reasonable request.
I donÕt do smileys.
--
I donÕt want to wine and dine and date you once or twice.
I want to hold you now. I just want to spend the night.
You tell me a better plan. Baby, IÕm not a patient man.
-- Jimmy Lafave, the romantic troubadour.
===
Subject: Re: logic of the Cantorian followers mind
Please translate the latin.
Quidquid latine dictum sit, altum viditur.
Bob Kolker
===
Subject: Re: logic of the Cantorian followers mind
> Please translate the latin.
> Quidquid latine dictum sit, altum viditur.
Whatsoever is said in Latin, is seen as high(?).
===
Subject: Re: logic of the Cantorian followers mind
>> Please translate the latin.
>> Quidquid latine dictum sit, altum viditur.
>Whatsoever is said in Latin, is seen as high(?).
High, or deep (go \figure). In this case, a good translation is
profound.
BTW, for the OP, a Google search gets lots of hits for this.
--
---------------------------
| BBB b Barbara at LivingHistory stop co stop uk
| B B aa rrr b |
| BBB a a r bbb | Quidquid latine dictum sit,
| B B a a r b b | altum viditur.
| BBB aa a r bbb |
-----------------------------
===
Subject: Re: logic of the Cantorian followers mind
:> : Which real numbers canÕt be \finitely \
described?
:> Most of them.
: Name one.
ÔbobÕ. :)
I am not sure that Ônamingand \
ÔdescribingÕ
are at all related in this case. I can name some
indescribable number ÔbobÕ, but that does not \
help you
describe or even identify that number, unless he is wearing
his name tag.
So I claim that Ôbobis a indescribable number. \
I do not
claim that Ôbobis a description of the \
indescribable,
just the name.
Of course there are only countably in\finite names,
so there are unnameable real numbers as well. :)
Stephen
===
Subject: Re: logic of the Cantorian followers mind
<co0h2q$poj$2@msunews.cl.msu.edu>
<41Qod.89279$T02.75468@twister.rdc-kc.rr.com>
<q2Qod.452553$D%.116021@attbi_s51>
<ymQod.89282$T02.74358@twister.rdc-kc.rr.com>
<co0kpi$t2i$1@msunews.cl.msu.edu>
<87act7fyz1.fsf@phiwumbda.org>
<co2gcl$2hp6$1@msunews.cl.msu.edu>
Discussion, linux)
> :> :> : Which real numbers canÕt be \finitely \
described?
> :> :> Most of them.
> : Name one.
> ÔbobÕ. :)
> I am not sure that Ônamingand \
ÔdescribingÕ
> are at all related in this case.
Well, probably neither of us should be spending so much time
on a
throwaway line, but...
IÕd say that the plain English interpretation of name as in
name
one is: specify one. Giving a name for which I donÕt know \
the
referent surely wouldnÕt count as satisfying my demand that
you name
one.
--
Jesse F. Hughes
ThatÕs whatÕs annoying about Usenet as some \
loser will state
a case,
get their ass kicked, but STILL keep coming back as if nothing
happened. -- James Harris explains his strategy.
===
Subject: Re: logic of the Cantorian followers mind
>> : Which real numbers canÕt be \finitely \
described?
>> Most of them.
>Name one.
x.
Lee Rudolph
===
Subject: Re: logic of the Cantorian followers mind
<co0h2q$poj$2@msunews.cl.msu.edu>
<41Qod.89279$T02.75468@twister.rdc-kc.rr.com>
<q2Qod.452553$D%.116021@attbi_s51>
<ymQod.89282$T02.74358@twister.rdc-kc.rr.com>
<co0kpi$t2i$1@msunews.cl.msu.edu>
<87act7fyz1.fsf@phiwumbda.org>
<co1ph4$jgg$1@panix2.panix.com>
Discussion, linux)
> : Which real numbers canÕt be \finitely \
described?
> Most of them.
>>Name one.
> x.
x has a \finite description. It is the least real number
greater than
or equal to x.
Nice try.
--
Jesse F. Hughes
I have written many words to sci.math, some of them are not
even
meaningless. --Ross Finlayson
===
Subject: Re: logic of the Cantorian followers mind
> : Which real numbers canÕt be \finitely \
described?
> Most of them.
>Name one.
>> x.
> x has a \finite description. It is the least real number
greater than
> or equal to x.
> Nice try.
Berry nice try.
--
Dave Seaman
Judge YohnÕs mistakes revealed in Mumia Abu-Jamal ruling.
<http://www.commoncouragepress.com/index.cfm?action=book&
bookid=228>
===
Subject: Re: logic of the Cantorian followers mind
> :> :> The mainstream accepts the existence of objects that
canÕt be
> :> \finitely
> :> :> described.
> :> : Exactly what objects are those?
> :> Most of the real numbers. Most of the sets of integers.
> :> Most of the languages over any alphabet.
> :>>ThatÕs a \finite description of those \
things.
> :> :> No, those are \finite descriptions of the *class* of
each of those
> :> things. But by CantorÕs argument, those classes each
contain elements
> :> that are not \finitely describable.
> :> --
> :> Josh Purinton
> : Which real numbers canÕt be \finitely \
described?
> Most of them.
> Stephen
Most of the real numbers might include 3/4 and pi. Are you
wrong, incabable of answering, or what?
===
Subject: Re: logic of the Cantorian followers mind
:> :> :> The mainstream accepts the existence of objects that
canÕt be
:> :> \finitely
:> :> :> described.
:> :> : Exactly what objects are those?
:> :> Most of the real numbers. Most of the sets of integers.
:> :> Most of the languages over any alphabet.
:> :>>ThatÕs a \finite description of those \
things.
:> :>
:> :> No, those are \finite descriptions of the *class* of each
of those
:> :> things. But by CantorÕs argument, those classes each
contain
elements
:> :> that are not \finitely describable.
:> :> --
:> :> Josh Purinton
:> : Which real numbers canÕt be \finitely \
described?
:> Most of them.
:> Stephen
: Most of the real numbers might include 3/4 and pi.
What is that suppose to mean? 3/4 and pi are describable,
so they are clearly not in the set of indescribable numbers.
There is no Ômightabout it.
: Are you
: wrong, incabable of answering, or what?
Most real numbers are indescribable. It is a simple
consequence of the de\finitions. The set of descriptions
is countably in\finite. The set of real numbers is
not countably in\finite.
Are you capable of understanding that? There are more
real numbers than descriptions. There are so many more
real numbers than descriptions that by any reasonable
de\fintion of ÔmostÕ, most real \
numbers are not describable.
Stephen
===
Subject: Re: logic of the Cantorian followers mind
<snip> Most real numbers are indescribable. It is a simple
> consequence of the de\finitions. The set of descriptions
> is countably in\finite. The set of real numbers is
> not countably in\finite.
> Are you capable of understanding that? There are more
> real numbers than descriptions. There are so many more
> real numbers than descriptions that by any reasonable
> de\fintion of ÔmostÕ, most real \
numbers are not describable.
> Stephen
Which numbers canÕt be described by a point on a number \
line?
===
Subject: Re: logic of the Cantorian followers mind
: <snip>
:> Most real numbers are indescribable. It is a simple
:> consequence of the de\finitions. The set of descriptions
:> is countably in\finite. The set of real numbers is
:> not countably in\finite.
:> Are you capable of understanding that? There are more
:> real numbers than descriptions. There are so many more
:> real numbers than descriptions that by any reasonable
:> de\fintion of ÔmostÕ, most real \
numbers are not describable.
:> Stephen
: Which numbers canÕt be described by a point on a number
line?
How do you describe pi, or any number for that matter, using
a number
line?
Stephen
===
Subject: Re: logic of the Cantorian followers mind
> : <snip> :> Most real numbers are indescribable. It is a
simple
> :> consequence of the de\finitions. The set of descriptions
> :> is countably in\finite. The set of real numbers is
> :> not countably in\finite.
> :> :> Are you capable of understanding that? There are more
> :> real numbers than descriptions. There are so many more
> :> real numbers than descriptions that by any reasonable
> :> de\fintion of ÔmostÕ, most real \
numbers are not
describable.
> :> :> Stephen
> : Which numbers canÕt be described by a point on a number
line?
> How do you describe pi, or any number for that matter,
using a number
> line?
> Stephen
The same way most people do.
===
Subject: Re: logic of the Cantorian followers mind
:> : <snip>
:> :> Most real numbers are indescribable. It is a simple
:> :> consequence of the de\finitions. The set of descriptions
:> :> is countably in\finite. The set of real numbers is
:> :> not countably in\finite.
:> :>
:> :> Are you capable of understanding that? There are more
:> :> real numbers than descriptions. There are so many more
:> :> real numbers than descriptions that by any reasonable
:> :> de\fintion of ÔmostÕ, most \
real numbers are not
describable.
:> :>
:> :> Stephen
:> : Which numbers canÕt be described by a point on a number
line?
:> How do you describe pi, or any number for that matter,
using a number
:> line?
:> Stephen
: The same way most people do.
I am quite sure most people have never tried to describe pi
using a number line. So I guess that means that you do not
know how to describe pi using a number line.
Stephen
===
Subject: Re: logic of the Cantorian followers mind
> :> :> : <snip> :> :> Most real numbers are indescribable.
It is a simple
> :> :> consequence of the de\finitions. The set of \
descriptions
> :> :> is countably in\finite. The set of real numbers is
> :> :> not countably in\finite.
> :> :> :> :> Are you capable of understanding that? There
are more
> :> :> real numbers than descriptions. There are so many more
> :> :> real numbers than descriptions that by any reasonable
> :> :> de\fintion of ÔmostÕ, most \
real numbers are not
describable.
> :> :> :> :> Stephen
> :> :> : Which numbers canÕt be described by a point on a
number line?
> :> :> How do you describe pi, or any number for that
matter, using a number
> :> line?
> :> :> Stephen
> : The same way most people do.
> I am quite sure most people have never tried to describe pi
> using a number line. So I guess that means that you do not
> know how to describe pi using a number line.
> Stephen
I guess that means you donÕt know how to do it. Otherwise
you wouldnÕt need for me to tell you these things.
===
Subject: Re: logic of the Cantorian followers mind
In sci.logic, Poker Joker
<Poker@wi.rr.com>
<7FTod.89499$T02.4671@twister.rdc-kc.rr.com>:
>> :>> :> : <snip>> :> :> Most real numbers are
indescribable. It is a simple
>> :> :> consequence of the de\finitions. The set of
descriptions
>> :> :> is countably in\finite. The set of real numbers is
>> :> :> not countably in\finite.
>> :> :>> :> :> Are you capable of understanding that? There
are more
>> :> :> real numbers than descriptions. There are so many
more
>> :> :> real numbers than descriptions that by any reasonable
>> :> :> de\fintion of ÔmostÕ, most \
real numbers are not
describable.
>> :> :>> :> :> Stephen
>> :>> :> : Which numbers canÕt be described by a point on a
number line?
>> :>> :> How do you describe pi, or any number for that
matter, using a number
>> :> line?
>> :>> :> Stephen
>> : The same way most people do.
>> I am quite sure most people have never tried to describe pi
>> using a number line. So I guess that means that you do not
>> know how to describe pi using a number line.
>> Stephen
> I guess that means you donÕt know how to do it. Otherwise
> you wouldnÕt need for me to tell you these things.
IÕm not sure one can describe an arbitrary point on the line
anyway,
unless it happens to be in one of the following sets.
[1] Q.
[2] A known transcendental, such as e, pi, log(2).
[3] The solution of an equation, usually (but not always!)
polynomial
in nature.
[4] Any arithmetic combination of the above.
[5] The limit point of certain sequences or series, where the
terms
can be computed easily (e.g., sum(i=1,+oo) (1/i!) yields e;
sum(i=1,+oo) (1/i^2) yields pi^2/6, IIRC). Note that an
in\finite
decimal expansion \fits into this category; the number
.012345678910111213141516171819... is one example of many.
This territory becomes fairly murky quickly, as you might well
imagine.
--
#191, ewill3@earthlink.net
ItÕs still legal to go .sigless.
===
Subject: Re: logic of the Cantorian followers mind
><snipped>> The
>> mainstream accepts the existence of objects that canÕt be
\finitely
>> described.
>Exactly what objects are those?
Well, consider the set of uncomputable real numbers. Now that
set is an
acceptable mainstream object, and it can be \finitely
described. So it
presumably exists in the sense youÕre intending. Well, in \
set
theory,
if
a set exists then each of its individual elements also
exists, right?
Therefore particular uncomputable real numbers exist, right?
But no
particular uncomputable real number can be \finitely described.
--
---------------------------
| BBB b Barbara at LivingHistory stop co stop uk
| B B aa rrr b |
| BBB a a r bbb | Quidquid latine dictum sit,
| B B a a r b b | altum viditur.
| BBB aa a r bbb |
-----------------------------
===
Subject: Re: logic of the Cantorian followers mind
<6yPod.89276$T02.37355@twister.rdc-kc.rr.com>
<co0hth$b9o$1@lust.ihug.co.nz>
Discussion, linux)
>><snipped> The
> mainstream accepts the existence of objects that canÕt be
\finitely
> described.
>>Exactly what objects are those?
> Well, consider the set of uncomputable real numbers. Now
that set is an
> acceptable mainstream object, and it can be \finitely
described. So it
> presumably exists in the sense youÕre intending. Well, in
set theory,
if
> a set exists then each of its individual elements also
exists, right?
> Therefore particular uncomputable real numbers exist,
right? But no
> particular uncomputable real number can be \finitely
described.
Depends on what you mean by \finitely described. \
ChaitinÕs
Omega can
be \finitely described, canÕt it?
--
Jesse F. Hughes
I donÕt know if you noticed but I had a tremendous drop in
con\fidence
concomittant [sic] with a dramatic grip of existential crisis.
--- James S. Harris even has better diseases than you
===
Subject: Re: logic of the Cantorian followers mind
><snipped>> The
>> mainstream accepts the existence of objects that canÕt be
\finitely
>> described.
>Exactly what objects are those?
>> Well, consider the set of uncomputable real numbers. Now
that set is an
>> acceptable mainstream object, and it can be \finitely
described. So it
>> presumably exists in the sense youÕre intending. Well, in
set
theory, if
>> a set exists then each of its individual elements also
exists, right?
>> Therefore particular uncomputable real numbers exist,
right? But no
>> particular uncomputable real number can be \finitely
described.
>Depends on what you mean by \finitely described. \
ChaitinÕs
Omega can
>be \finitely described, canÕt it?
YouÕre right, that is a ßawed argument (as \
IÕve already
admitted).
The cardinalities argument is the appropriate one to use:
there are
\finitely-de\fined sets (such as R) which have more \
elements than
there are
possible labels for.
--
---------------------------
| BBB b Barbara at LivingHistory stop co stop uk
| B B aa rrr b |
| BBB a a r bbb | Quidquid latine dictum sit,
| B B a a r b b | altum viditur.
| BBB aa a r bbb |
-----------------------------
===
Subject: Re: logic of the Cantorian followers mind
[snipped]
>>Depends on what you mean by \finitely described. \
ChaitinÕs
Omega can
>>be \finitely described, canÕt it?
> YouÕre right, that is a ßawed argument (as \
IÕve already
admitted).
By convention, you must be a troll.
===
Subject: Re: logic of the Cantorian followers mind
>><snipped> The
> mainstream accepts the existence of objects that canÕt be
\finitely
> described.
>>Exactly what objects are those?
> Well, consider the set of uncomputable real numbers. Now
that set is an
> acceptable mainstream object, and it can be \finitely
described. So it
> presumably exists in the sense youÕre intending. Well, in
set theory,
> if
> a set exists then each of its individual elements also
exists, right?
> Therefore particular uncomputable real numbers exist,
right? But no
> particular uncomputable real number can be \finitely
described.
Of course I pointed out you were wrong. You didnÕt know what
you were talking about. Subsequently you called me a troll.
Why do
you bother to even post. I didnÕt call you a troll or any
other name
for that matter when you posted the bullshit above.
===
Subject: Re: logic of the Cantorian followers mind
<6yPod.89276$T02.37355@twister.rdc-kc.rr.com>
<co0hth$b9o$1@lust.ihug.co.nz>
<A6Uod.89564$T02.84389@twister.rdc-kc.rr.com>
posting-account=Qiuj5AwAAACmGnmS12qcvqA9IXzD0s4L
Agreed, this is poorly stated:
> But no
> particular uncomputable real number can be \finitely
described.
Barb is swapping between label and encapsulate when she uses
the
word describe.
Herc
===
Subject: Re: logic of the Cantorian followers mind
> But no
> particular uncomputable real number can be \finitely
described.
DoesnÕt Chaitin describe a uncomputable number?
===
Subject: Re: logic of the Cantorian followers mind
>> But no
>> particular uncomputable real number can be \finitely
described.
>DoesnÕt Chaitin describe a uncomputable number?
ThatÕs a good point. My initial reasoning was overly complex
to begin with.
HereÕs a simpler version that handles \
ChaitinÕs omega too:
Consider the set of real numbers. That set is an acceptable
mainstream
object, and it can be \finitely described. So it presumably
exists in
the
sense youÕre intending. Well, in set theory, if a set exists
then each of
its particular elements also exists, right? Therefore every
particular real
number exists. But, the reals are uncountable and the set of
possible
\finite descriptions (over a \finite alphabet) is \
countable, so
most
particular reals do not have any \finite description -- there
are not enough
\finite descriptions to go around.
--
---------------------------
| BBB b Barbara at LivingHistory stop co stop uk
| B B aa rrr b |
| BBB a a r bbb | Quidquid latine dictum sit,
| B B a a r b b | altum viditur.
| BBB aa a r bbb |
-----------------------------
===
Subject: Re: logic of the Cantorian followers mind
> But no
> particular uncomputable real number can be \finitely
described.
>>DoesnÕt Chaitin describe a uncomputable number?
> ThatÕs a good point. My initial reasoning was overly
complex to begin
> with.
> HereÕs a simpler version that handles \
ChaitinÕs omega too:
> Consider the set of real numbers. That set is an acceptable
mainstream
> object, and it can be \finitely described. So it presumably
exists in
> the
> sense youÕre intending. Well, in set theory, if a set
exists then each
> of
> its particular elements also exists, right? Therefore every
particular
> real
> number exists. But, the reals are uncountable and the set
of possible
> \finite descriptions (over a \finite alphabet) is \
countable, so
most
> particular reals do not have any \finite description -- there
are not
> enough
> \finite descriptions to go around.
LetÕs start here:
CanÕt I put a point on a number line? Conceptually that \
point
can go
anywhere, canÕt it? Which reals canÕt be \
represented by that
number
line and point? Is that a \finite description?
Or how about this:
One of the reals. is a sentence that applies to each and
every real
number. What part of that sentence isnÕt \
\finite?
What about this:
Well, in set theory, if a set exists then each of its
particular elements
can be selected, right? Therefore we must be able to describe
it.
Hmmm....
ArenÕt all objects that are in existence in this \
\finite world
forced
into the world of \finiteness? OTOH - Objects that \
donÕt really
exist might not have a \finite description.
===
Subject: Re: logic of the Cantorian followers mind
> But no
> particular uncomputable real number can be \finitely
described.
>DoesnÕt Chaitin describe a uncomputable number?
> ThatÕs a good point. My initial reasoning was overly
complex to begin
> with.
> HereÕs a simpler version that handles \
ChaitinÕs omega too:
> Consider the set of real numbers. That set is an acceptable
mainstream
> object, and it can be \finitely described. So it presumably
exists
in
> the
> sense youÕre intending. Well, in set theory, if a set
exists then each
> of
> its particular elements also exists, right? Therefore every
particular
> real
> number exists. But, the reals are uncountable and the set
of possible
> \finite descriptions (over a \finite alphabet) is \
countable, so
most
> particular reals do not have any \finite description -- there
are not
> enough
> \finite descriptions to go around.
> LetÕs start here:
> CanÕt I put a point on a number line? Conceptually that
point can go
> anywhere, canÕt it? Which reals canÕt be \
represented by
that number
> line and point? Is that a \finite description?
> Or how about this:
> One of the reals. is a sentence that applies to each and
every real
> number. What part of that sentence isnÕt \
\finite?
> What about this:
> Well, in set theory, if a set exists then each of its
particular elements
> can be selected, right? Therefore we must be able to
describe it.
Yes, as a set. A real (between 0 an 1) is represented by a
function from
N, the natural numbers, to the set {0,1}. We interpret that
as the
binary expansion of a real. A function is a particular type
of set. It
is not necessary in the de\finition of a function for the
function to be
able to be DESCRIBE (which I interpret as, say, generated by
a \finite
algorithm).
Take the real number
.0101010101010101010101001111011110011101010100111...
If you ask me, what is the 47-th place, I might say, 1. And
if you ask
me the 348784845784-th place, I say 0.
For any n, I will give you back a 0 or 1. And if you give the
the same n
repeatedly, each time you give me a particular n, IÕll \
always
give you
That is what makes f a FUNCTION. It is NOT REQUIRED (sorry
for the
shouting, IÕm getting agitated) to be able to describe f \
with
an
algorithm or a process. ItÕs only necessary that every time
you ask
whatÕs the 47-th digit, I give you the same answer.
===
Subject: Re: logic of the Cantorian followers mind
<6yPod.89276$T02.37355@twister.rdc-kc.rr.com>
<co0hth$b9o$1@lust.ihug.co.nz>
<1pQod.89283$T02.68184@twister.rdc-kc.rr.com>
<co0lft$g0j$1@lust.ihug.co.nz>
posting-account=Qiuj5AwAAACmGnmS12qcvqA9IXzD0s4L
>particular elements also exists, right? Therefore every
particular
real
>number exists. But, the reals are uncountable and the set of
possible
>\finite descriptions (over a \finite alphabet) is \
countable, so
most
>particular reals do not have any \finite description -- there
are not
enough
>\finite descriptions to go around.
marvellous! worthy of men in suits to go doorknocking with.
reminds
me
of the morman bible page I read where a mystical spirit told
the
mormans
to go and tell other people, and to tell them to tell more
people!
wow, an unlistable in\finite universe of undescribable numbers,
all
because
you can ßip the digit sequence down a list and extrapolate it
(wrongly) to in\finity.
did you ever notice the bable in undescribable?
Herc
===
Subject: Re: logic of the Cantorian followers mind
>><snipped> The
> mainstream accepts the existence of objects that canÕt be
\finitely
> described.
>>Exactly what objects are those?
> Well, consider the set of uncomputable real numbers. Now
that set is an
> acceptable mainstream object, and it can be \finitely
described. So it
> presumably exists in the sense youÕre intending. Well, in
set theory,
> if
> a set exists then each of its individual elements also
exists, right?
> Therefore particular uncomputable real numbers exist,
right? But no
> particular uncomputable real number can be \finitely
described.
> --
> ---------------------------
> | BBB b Barbara at LivingHistory stop co stop
uk
> | B B aa rrr b |
> | BBB a a r bbb | Quidquid latine dictum sit,
> | B B a a r b b | altum viditur.
> | BBB aa a r bbb |
> -----------------------------
Not a bad description of objects you canÕt \
\finitely describe.
===
Subject: Re: Computer language and category theory
> Is there any work done one computer languages and category
theory?
[...]
> However, I donÕt like such animals. So I wonder if there \
is
some use
> of category theory or something else that have been used to
model
> language like constructs.
Any formalism for transition systems generalizes to something
involving
categories. Suppose G = (Q,s,M,P) is a grammar; i.e., s is in
Q,
Q is a set of variables, M is a monoid (the standard
formalism admits
only free monoids M = X*, where X is then deemed the
ÔalphabetÕ, but
everything works in the general case), and P is a subset of Q
x M[Q],
where M[Q] is the free extension of M over the set Q (note
then that
(X*)[Q] is just (X union Q)*).
A transition relation -> can be de\fined over M[Q] recursively
by the
following conditions:
(a) a -> a, for any a in M[Q]
(b) if a -> b, b -> c, then a -> c, where a, b, c are in M[Q]
(c) if (q,b) is in P then aqc -> abc, where a, c are in M[Q]
This has the following properties:
* Let [a] = { m in M: a -> m }
then
[m] = {m} for all m in M
Context freeness: [ab] = [a][b] for all a, b in M[Q]
[q] = union {[a]: (q,a) in P}
* if a -> b, c -> d then ac -> bd
* the set { q = [q]: q in Q } is the least solution to the
set-theoretic system de\fined from the grammar
(i.e., where a rule q -> xry becomes the inequality
[q] superseet of {x}[r]{y} for q,r in Q, x,y in M
similarly for the other rules).
The morphisms are de\fined recursively by:
I: a -> a
if f: a -> b, g: b -> c then gf: a -> c
(q,b): aqc -> abc
Alternatively, one can restrict the last family of morphisms
only to the following: (q,b): q -> b and then add another
family
of the form
if f: a -> b, g: c -> d then f x g: ac -> bd
given the property just cited above.
When the grammar is not cyclic then s is an initial object.
All the elements m of M are terminal objects. The language
L(G) is just [s] which is the set of elements m of M for
which morphisms f: s -> m exist. Each morphism corresponds
roughly to a derivation sequence.
===
Subject: General Harmonic analysis question
I know there doesnÕt exist a general theory for harmonic
analysis on
non-locally compact groups. What useful applications would a
theory for
harmonic analysis on non-locally compact groups have? Are
there real world
practical applications? In particular, does anyone need to
study continuous
functions from non-compact groups to the real line or
anything similar?
In order to create a theory for non-locally compact groups,
would there have
to be a Haar integral analog, a Peter-Weyl analog, and
Plancherel theorem
analog?
Isaac
===
Subject: Re: help! solving matrix equations ...
>realistic problems, I forgot to limit that y cannot be 0.
and in fact, I
>want ||w||=1... the norm of w = 1.
>Sorry I forgot this practicality constraint...
>Under this new added condition, what can be a good solution?
Ah, that changes things considerably. So now you want to
minimize || P a - C w || subject to ||w|| = 1. Using a
Lagrange
multiplier, we can write the Lagrangian as
F(a,w,lambda) = (P a - C w)^T (P a - C w) - lambda (w^T w - 1)
We look for a stationary point, where the gradients with
respect to
a and w are both 0. I get
C^T P a = (C^T C - lambda) w
P^T P a = - P^T C w
Assuming P^T P is invertible, you want a = - (P^T P)^(-1) P^T
C w, and
then
C^T (I + P (P^T P)^(-1) P^T) C w = lambda w
i.e. w should be an eigenvector of C^T (I + P (P^T P)^(-1)
P^T) C
for eigenvalue lambda. Moreover, I then get
||P a - C w||^2 = lambda ||w||^2
so you want to take a normalized eigenvector for the lowest
eigenvalue.
Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
===
Subject: Re: help! solving matrix equations ...
>realistic problems, I forgot to limit that y cannot be 0.
and in fact, I
>want ||w||=1... the norm of w = 1.
>Sorry I forgot this practicality constraint...
>Under this new added condition, what can be a good solution?
> Ah, that changes things considerably. So now you want to
> minimize || P a - C w || subject to ||w|| = 1. Using a
Lagrange
> multiplier, we can write the Lagrangian as
> F(a,w,lambda) = (P a - C w)^T (P a - C w) - lambda (w^T w -
1)
> We look for a stationary point, where the gradients with
respect to
> a and w are both 0. I get
> C^T P a = (C^T C - lambda) w
> P^T P a = - P^T C w
> Assuming P^T P is invertible, you want a = - (P^T P)^(-1)
P^T C w, and
> then
> C^T (I + P (P^T P)^(-1) P^T) C w = lambda w
> i.e. w should be an eigenvector of C^T (I + P (P^T P)^(-1)
P^T) C
> for eigenvalue lambda. Moreover, I then get
> ||P a - C w||^2 = lambda ||w||^2
> so you want to take a normalized eigenvector for the lowest
> eigenvalue.
> Robert Israel israel@math.ubc.ca
> Department of Mathematics http://www.math.ubc.ca/~israel
> University of British Columbia Vancouver, BC, Canada
Just in case your silence is a result of you being completely
overwhelmed, IÕd be happy to contribute an informa \
discussion
of the
absolute basics, starting with Robert IsraelÕs original
suggestion to
solve the equation
[P|C]*x=nullvector
Let me know.
Reinhard
===
Subject: Re: Why Do Americans Call It Math?
> I used to say Maths, as a Brit, I still do sometimes in the
UK, but I
gave
> putting the s on the end makes it awkward to lisp out. Math
avoids the
> horrible triple consonant ÔthsÕ
> ThatÕs a double consonant, and occurs in lots of plurals.
ItÕs easy
enough.
Indeed; one can only assume that he doesnÕt take baths.
And I bet he doesnÕt have any diphthongs in his catchphrase!
Phil
--
... one Marine noticed one of the prisoners was still
breathing. A Marine
can be heard saying on the pool footage provided to Reuters
Television:
HeÕs fucking faking heÕs dead. He faking \
heÕs fucking dead.
The Marine then raises his riße and \fires into the \
manÕs head.
The pictures are too graphic for us to broadcast, Sites said.
===
Subject: Re: Why Do Americans Call It Math?
>
> I used to say Maths, as a Brit, I still do sometimes in the
UK, but I
gave
> putting the s on the end makes it awkward to lisp out. Math
avoids
the
> horrible triple consonant ÔthsÕ
>
> ThatÕs a double consonant, and occurs in lots of plurals.
ItÕs easy
enough.
> Indeed; one can only assume that he doesnÕt take baths.
> And I bet he doesnÕt have any diphthongs in his \
catchphrase!
But he does play to his strengths.
Paul
--
Hanging on in quiet desperation is the English way.
The time is gone, the song is over.
Thought IÕd something more to say.
===
Subject: Re: Why Do Americans Call It Math?
>
>
> I used to say Maths, as a Brit, I still do sometimes in the
UK, but
I gave
> putting the s on the end makes it awkward to lisp out. Math
avoids
the
> horrible triple consonant ÔthsÕ
>
> ThatÕs a double consonant, and occurs in lots of plurals.
ItÕs easy
enough.
When I was in fourth grade, our teacher was saying that is was
uncommon to have a word with three consecutive consonants. I
instantly raised my hand to say I knew a word with \five
consecutive
consonants: thousandths. She nicely informed that was only
four,
because th was a single sound.
David Ames
>
> Indeed; one can only assume that he doesnÕt take baths.
>
> And I bet he doesnÕt have any diphthongs in his \
catchphrase!
> But he does play to his strengths.
> Paul
===
Subject: Re: Why Do Americans Call It Math?
> horrible triple consonant ÔthsÕ
>
> ThatÕs a double consonant, and occurs in lots of plurals.
ItÕs easy
enough.
>> Indeed; one can only assume that he doesnÕt take baths.
>> And I bet he doesnÕt have any diphthongs in his
catchphrase!
>But he does play to his strengths.
I guess someone will have to add a postscript about his
hardscrabble
offspringÕs jockstrap heading downstream to his birthplace,
\first
lengthwise then in a corkscrew manner. I saw it in a truly
earthshaking \filmstrip.
===
Subject: Re: Why Do Americans Call It Math?
> horrible triple consonant ÔthsÕ
>
> ThatÕs a double consonant, and occurs in lots of plurals.
ItÕs easy
enough.
>>
>> Indeed; one can only assume that he doesnÕt take baths.
>>
>> And I bet he doesnÕt have any diphthongs in his
catchphrase!
>But he does play to his strengths.
> I guess someone will have to add a postscript about his
hardscrabble
> offspringÕs jockstrap heading downstream to his \
birthplace,
\first
> lengthwise then in a corkscrew manner. I saw it in a truly
> earthshaking \filmstrip.
How abstract. I assume itÕs set in his birthplace
Christchurch, and
starring Frenchmen in their nightclothes, I assume? Such
heartthrobs
tug at my heartstrings.
===
Subject: Re: Why Do Americans Call It Math?
>> And I bet he doesnÕt have any diphthongs in his
catchphrase!
>But he does play to his strengths.
> I guess someone will have to add a postscript about his
hardscrabble
> offspringÕs jockstrap heading downstream to his \
birthplace,
\first
> lengthwise then in a corkscrew manner. I saw it in a truly
> earthshaking \filmstrip.
> How abstract. I assume itÕs set in his birthplace
Christchurch, and
> starring Frenchmen in their nightclothes, I assume? Such
heartthrobs
> tug at my heartstrings.
I wish this thread was in danish. I would have loved to use
the word angstskrig.
Asger.
===
Subject: Re: Why Do Americans Call It Math?
>> And I bet he doesnÕt have any diphthongs in his
catchphrase!
>But he does play to his strengths.
> I guess someone will have to add a postscript about his
hardscrabble
> offspringÕs jockstrap heading downstream to his \
birthplace,
\first
> lengthwise then in a corkscrew manner. I saw it in a truly
> earthshaking \filmstrip.
> How abstract. I assume itÕs set in his birthplace
Christchurch, and
> starring Frenchmen in their nightclothes, I assume? Such
heartthrobs
> tug at my heartstrings.
> I wish this thread was in danish. I would have loved to use
> the word angstskrig.
Use it often enough in English contexts and thereÕs a chance
it may be
accepted in the language. English is notorious for taking
words from
other languages and, usually, mispronouncing them. Recent
examples
loaded with consonant clusters include ersatz, perestroika and
smorgasbord. Note that we donÕt hold with those funny
mutilations
to the vowels which foreigners seem to use. 8-)
Paul
--
Hanging on in quiet desperation is the English way.
The time is gone, the song is over.
Thought IÕd something more to say.
===
Subject: Re: Why Do Americans Call It Math?
spose I better answer really...
some respondent was right, I \find \
Ôbathshard to say. At
least Thistle has
an Ôiseprating the h and the s.
Since this is a Math newsgroup, IÕll end it now.
Have a 1004 digit prime twin for bedtime,
1^1004+189770491,93. Systematic
search of \first 1600 due any time real soon now (as Bill G.
used to say)
gould luck and gould day
whatÕs a dipthong...?
donÕt bother
Richard Miller
>> And I bet he doesnÕt have any diphthongs in his
catchphrase!
>But he does play to his strengths.
> I guess someone will have to add a postscript about his
hardscrabble
> offspringÕs jockstrap heading downstream to his \
birthplace,
\first
> lengthwise then in a corkscrew manner. I saw it in a truly
> earthshaking \filmstrip.
> How abstract. I assume itÕs set in his birthplace
Christchurch, and
> starring Frenchmen in their nightclothes, I assume? Such
heartthrobs
> tug at my heartstrings.
> I wish this thread was in danish. I would have loved to use
> the word angstskrig.
> Asger.
===
Subject: Re: Principle = Principal Curvatures of a
Hypersurface
In R3 the prototypical situation is that of the surface z =
Axx/2 + Bxy
+ Cyy/2 at O = (x, y) = (0, 0).
According to whether BB - AC > 0, =0 or < 0 it is a hyperbolic
paraboloid, a parabolic cylinder or an elliptic paraboloid.
The principal curvatures at O are the eigenvalues of the
symmetric 2x2
matrix belonging in the standard way to this quadratic form. A
transformation to principal axes yields z = \
L.xÕxÕ/2 +
M.yÕyÕ/2. The
coef\ficients L and M are the pricipal curvatures.
In Rn one can play exactly the same game: Establish a
Cartesian
coordinate system P-X1-X2-...-Xn at point P of the
hypersurface S with
axis Xn along the normal to S and axes X1, X2, ..., X(n-1) in
the
tangent hyperplane to S at P.
In speci\fic cases this change of coordinate system may be easy
or
dif\ficult. No general theory exists to tell this in advance.
If S is indeed C2-smooth at P then S is representable in
these local
coordinates by a function xn = f(x1, x2,...,x(n-1)).
Its Taylor series expansion starts with a quadratic form. The
coef\ficients are the pure and mixed 2nd-order partial
derivatives of f.
The eigenvalues of the symmetric matrix belonging to this
form are the
principal curvatures.
A transformation to principal axes yields xn = \
L1.x1Õ.x1/ 2
+ ... +
Ln-1.x(n-1)Õ.x(n-1)/ 2 + o(x^2).
The answers to your questions: (1) no. (2) no in general; yes
only after
transformation to principal axes.
(3) For n > 4: no, even if the tangent hyperplane is not
parallel to the
original Xn axis; one has to solve an n-th-degree polynomial
equation to
\find the eigenvalues.
Johan E. Mebius
>HereÕs one that has me stumped ...
>IÕm trying to \find the n-1 principle curvatures \
of a
hypersurface
>f(x_1,x_2...x_n) in an n-dimensional Hilbert space.
>How do I go about doing this? All the texts I have referred
to only
>consider surfaces in R3 with simple (not to mention obvious)
>parametrizations.
>Note that in my case, I dont have a handy parametrization of
the form
>x_j=x_j(p_1,p_2...p_n-1) , j=1,n (1)
>where p_1,p_2 ... p_n-1 are the n-1 parameters
characterizing the
>n-dimensional hypersurface.
>My questions boil down to
>(1) Is there a general method of determining a
parametrization of the
>form (1) for any given hypersurface?
>(2) Are the principle curvatures then merely the vector
norms of the
>diagonal elements of the rank n-1 tensor of 2nd derivatives
of the
>position vectors de\fined by (1)?
>(3) Can the principle curvatures be determined without
recourse to
>such a parametrization?
>Lets assume that the hyperfurface f has continuous 1st and
2nd partial
>derivatives w.r.t each of x_1,x_2 ...x_n.
>-Sharat
===
Subject: Re: Principle = Principal Curvatures of a
Hypersurface
>(1) Is there a general method of determining a
parametrization of the
>form (1) for any given hypersurface?
I have no proper answers, trying to extrapolate from 3D, I
stand to
correction for any inaccuracy or error.
No, even in 3D, parameterization exploits symmetry for
simpli\fication
in particular cases of surfaces of revolution, helical
surfaces,
torses etc. Geodesics in hyperspace upto an arbitrary
parameter of
obliqueness using geodesic polar coordinates could be
considered... a
lot of this is from GR.
>(2) Are the principle curvatures then merely the vector
norms of the
>diagonal elements of the rank n-1 tensor of 2nd derivatives
of the
>position vectors de\fined by (1)?
>(3) Can the principle curvatures be determined without
recourse to
>such a parametrization?
In 3D, the product of principal curavatures comes out purely
from the
metric related to Tensor product R1212/g. Gauss Egregium
theorem
states that it is an isometric mapping invariant, a pure
class product
of \first fundamental form metric coef\ficients, \
second
fundamental form
coef\ficients L,M, and N get eliminated in \final \
result only of
highest
order _products_of curvatures, but not individual curvatures.
This is
also true in n-dimensional Riemannian geometry. Flatlanders
need not
know a priori how their land is bent or twisted in the
embedded
surrounding hyperspace, knowing the corresponding multiple
curvature
product hyper-invariant.
===
Subject: shapes and circle help
Hello
Are there any good methods to solve the 2 excercises below
relating
to shapes and circles
The wheel of a wheelbarrow rotates 60 times when it is pushed
a
distance of 50 m
calculate the radius of the wheel
and
the small circle has an areaof 16piecm2
the larger circle has a circumference of 18 pie cm
calculate the SHADED area
give your answer in terms of pie
for this you have to imagine a small circle in a big circle.
===
Subject: Re: shapes and circle help
> Are there any good methods to solve the 2 excercises below
relating
> to shapes and circles?
Yes. Read a few more examples. After learning the formulas for
perimeter and area of circles, watch how they are being
usefully
applied in those examples.
===
Subject: Root \finder XI
Root Finder xi.
by Jon Giffen.
It is found that the roots to the polynomial,
a[0]+a[1]t+a[2]t^2+...+a[n]t^n where
T=(t,t^2,t^3,..,t^n) and
N=(a[1],a[2],a[3],...,a[n]) are given by,
(D*N)|S|^2 D + (S*N)|D|^2 S
T = (-a[0]){---------------------------}
(D*N)^2|S|^2 + (S*N)^2|D|^2
where
D=(1,2,3,...,n) S=(a[1],2a[2],3a[3],...,na[n])
Solve the 6th degree polynomial,
t^6 + t - 10=0
a[0] = -10 N=(1,0,0,0,0,1) D=(1,2,3,4,5,6) S=(1,0,0,0,0,6)
D*N=7 S*N=7 |D|^2 = 91 |S|^2 = 37
37(1,2,3,4,5,6)+91(1,0,0,0,0,6)
T = (10)-------------------------------
7(37+91)
(1280,740,1110,1480,1850,7680)
= ------------------------------ = (t,t^2,t^3,t^4,t^5,t^6)
896
since t and t^6 need only be considered,
t = 1280/896 = 10/7
t^6 = 7680/896 = 60/7
60/7 + (60/7)^(1/6) - 10 = 0.002 which is almost zero.
(10/7)^6 + 10/7 - 10 = -0.07 which is also close
Solve the 6th degree polynomial,
t^6 - t - 10=0
t = -(60/7)^(1/6) from the prior example
f(t) = t^7 + t^3 + t - 20 = 0
a[0] = -20 N=(1,0,1,0,0,0,1) D=(1,2,3,4,5,6,7)
S=(1,0,3,0,0,0,7)
D*N = 11 S*N = 11 |S|^2 = 59 |D|^2 = 138
59(1,2,3,4,5,6,7)+138(1,0,3,0,0,0,7)
T = (20)------------------------------------
11(59+138)
t^7 = 27580/2167 t=1.4382 f(1.4382) = -2.8597
applying this to NewtonÕs Method,
1.4382 - (-2.857)/[7(1.4382^6)+3(1.4382^2) + 1] = 1.4795
f(1.4795)=0.2361
1.4795 - (0.2361)/[7(1.4795^6)+3(1.4795^2) + 1] = 1.4766
f(1.4766)=0.000092 ~ 0
f(t) = t^7 + t^3 + t + 20 = 0
t = -1.4766 from last example
f(t) = 2t^4 + 3t^3 + 2t^2 + t - 13 = 0
T*(1,2,3,2)-13 = 0 a[0]= -13
N=(1,2,3,2) D=(1,2,3,4) S=(1,4,9,8) |D|^2=30 |S|^2 = 162
D*N=22 S*N=52
13[22(162)(1,2,3,4)+52(30)(1,4,9,8)]
T = --------------------------------------
162(22^2) + 30(52^2)
t^4 = 347568/159520 = 2.1787 t = 1.2149 f(1.2149) = 0.9038
applying NewtonÕs Method,
1.2149 - 0.9038/[8(1.2149^3)+9(1.2149^2)+4(1.2149)+1] = 1.1879
f(1.1879) = 0.0213 ~ 0
f(t)=t^6 - t^5 + 4t^4 - 5t^3 + t^2 - t - 100 = 0
(-1,1,-5,4,-1,1)*T - 100 = 0
dividing the negatives from the positives,
(0,1,0,4,0,1)*T - (1,0,5,0,1,0)*T = 100
(0,1,0,4,0,1)*T - 100p = (1,0,5,0,1,0)*T + 100(1-p) = 0
p is some ratio
Applying the formula to each partition of N,
(D*N)|S|^2 D + (S*N)|D|^2 S
T = (-a[0]){---------------------------}
(D*N)^2|S|^2 + (S*N)^2|D|^2
where
D=(1,2,3,...,n) S=(a[1],2a[2],3a[3],...,na[n])
for (0,1,0,4,0,1)*T - 100p , a[0]=-100p
D*N=2+16+6=24 |S|^2=4+16^2+36=296 S*N=2+16+6=24
|D|^2=1+4+9+16+25+36=91
for (1,0,5,0,1,0)*T + 100(1-p) , a[0]=100(1-p)
D*N=1+15+5=21 |S|^2=1+15^2+25=251 S*N=1+15+5=21 |D|^2=91
296(1,2,3,4,5,6)+91(0,2,0,16,0,6)
(100p)----------------------------------
24(296+91)
251(1,2,3,4,5,6)+91(1,0,15,0,5,0)
=100(p-1)---------------------------------
21(251+91)
Taking the magnitude of both sides
|296(1,2,3,4,5,6)+91(0,2,0,16,0,6)|
=
|(296,774,888,2640,1480,2322)|=4003.36
|251(1,2,3,4,5,6)+91(1,0,15,0,5,0)|
=
|(342,502,2118,1004,1710,1506|=3324.91
4003.36 3324.91
p--------- = (p-1)-------
9288 7182
(0.9310)p = p-1
1 = (1-0.9310)p
p = 14.5
then
f(t)=t^6 - t^5 + 4t^4 - 5t^3 + t^2 - t - 100 = 0
t^6 = 1450/4 =362.5 t=2.67 f(2.67 )=239.38
t^5 = 858400/9288= 92.420 t=2.1263 f(2.1263)=-15.2200
Selecting -15.2200 for NewtonÕs Method,
f(t)=t^6 - t^5 + 4t^4 - 5t^3 + t^2 - t - 100 = 0
2.1263+15.22/
[6(2.1263^5)-5(2.1263^4)+16(2.1263^3)-15(2.1263^2)+2(2.1263)-
1]
t=2.1877 f(2.1877)= 1.3927
The development of this is given at,
http://mypeoplepc.com/members/jon8338/polynomial/id7.html
Jon Giffen
===
Subject: Root Finder 12
Root Finder 12
by Jon Giffen.
This solution was so simple that I couldnÕt believe it.
But I tried it, and it works.
It is found that the roots to the polynomial,
a[0]+a[1]t+a[2]t^2+...+a[n]t^n=0 where
T=(t,t^2,t^3,..,t^n) and
N=(a[1],a[2],a[3],...,a[n]) are given by,
-a[0]
T= -----D
D*N
D=(1,2,3,4,...,n)
Solve the 6th degree polynomial,
f(t)=t^6 + t - 10=0
a[0] = -10 N=(1,0,0,0,0,1) D=(1,2,3,4,5,6)
a[0]=-10 D*N=7
-a[0] 10
T = -----D = ---(1,2,3,4,5,6)
D*N 7
t =10/7 t=1.42835 f(1.42835)=-0.071
t^6=60/7 t=1.43056 f(1.43056)= 0.002
Solve the 49th degree polynomial,
f(t)=t^49 + t^16 + 40t^5 - 6000 = 0
a[0]=-6000 N=(1,0,0,0,40,0,..,1,0,...,1) D=(1,2,3,..,49)
D*N=1+200+16+49=266
-a[0] 6000
T = -----D = ------(1,2,3,..,49)
D*N 266
t^49=294000/266=1105.26 t=1.15375 f(1.15375)=-3575.99
Applying NewtonÕs Method,
t=1.15375-(-3575)/[49(1.15375^48)+16(1.15375^15)+200(1.15375)
^4]
=1.15375 again, so the answer must depend on distant decimal
places.
mixed signs, no pattern
f(t)=t^6 - t^5 + 4t^4 + 5t^3 + t^2 - t - 100 = 0
a[0]=-100 N=(-1,1,5,4,-1,1) D=(1,2,3,4,5,6)
D*N=-1+2+15+16-5+6=33
-a[0] 100
T = -----D = ---(1,2,3,4,5,6)
D*N 33
t^6=600/33=18.1818 t=1.62158 f(1.62158)=-43.0453 too low
t^5=500/33=15.1515 t=1.72223 f(1.72223)=-27.0815 too low
f^4=400/33=12.1212 t=1.86589 f(1.86589)= 2.16509 pretty close
Applying NewtonÕs Method,
1.86589-
2.16509
-------------------------------------------------------------
-----
6(1.86589^5)-5(1.86589^4)+16(1.86589^3)+15(1.86589^2)+2(
1.86589)-1
t=1.856637 f(1.856637)=0.0192
t^2 + 2t - 3 = 0
a[0]=-3 N=(2,1) D=(1,2) D*N=4
-a[0] 3
T = -----D =---(1,2) t^2=3/4 t=0.866 ~ 1
D*N 4
The lengthy development of this is given at,
http://mypeoplepc.com/members/jon8338/polynomial/id7.html
Jon Giffen
===
Subject: Root Finder 13
Root Finder 13
Jon Giffen
Another approach is considered, along with a possibility for
\finding the root to an In\finite Series.
It is discovered that the property of the nth degree
polynomial,
a[0]+a[1]t+a[2]t^2+a[3]t^3+...+a[n]t^n=0 where
N=(a[1],a[2],a[3],...,a[n])
T=(t.t^2.t^3,t^4,..., t^n )
Is,
|C|^2
|T|^2=(-a[0])^2{-------------------}
|N|^2|C|^2-(N*C)^2
where
C=(a[1],2a[2],3a[3],4a[4],...,na[n])
f(t)=t^6 - t^5 + 4t^4 + 5t^3 + t^2 - t - 100 = 0
a[0]=-100 N=(-1,1,5,4,-1,1) C=(-1,2,15,16,-5,6)
|C|^2=547 |N|^2=48 N*C=153
|N|^2 |C|^2 - (N*C)^2 = 48(547)-153^2=2847
|C|^2 D
T =(-a[0]){----------------------}^(1/2) ---
|N|^2 |C|^2 - (N*C)^2 |D| where
D=(1,2,3,4,5,6) and
100 547
T = -------- {-----}^(1/2) (1,2,3,4,5,6) =
4.594933(1,2,3,4,5,6)
91^(1/2) 2847
t^6=27.5696 t=1.738097
t^5=22.9746 t=1.871758
----------
t=1.856637 is the correct root
Notice that the root to a polynomial that is so long, that
it is virtually an in\finite Power Series; is found by using
the solution to the Geometric Series,
|C|^2
|T|^2=t^2+(t^2)^2+(t^2)^3+..+(t^2)^n=(-a[0])^2{--------------
----}
|N|^2|C|^2-(N*C)^2
adding 1 to both sides,
|C|^2
1+t^2+(t^2)^2+(t^2)^3+..+(t^2)^n=(-a[0])^2{------------------
}+1
|N|^2|C|^2-(N*C)^2
then the sum S=1/(1-t^2) but
|C|^2
S=(-a[0])^2{------------------}+1 (1-t^2)=1/S t^2=1-1/S and
|N|^2|C|^2-(N*C)^2
t ={1 - 1/S}^(1/2) where
N=(a[1],a[2],a[3],...,a[n])
T=(t,t^2,t^3,t^4,..., t^n )
C=(a[1],2a[2],3a[3],4a[4],...,na[n])
to the nth degree power series,
a[0]+a[1]t+a[2]t^2+a[3]t^3+...+a[n]t^n=0
Development
Suppose the polynomial,
a[0]+a[1]t+a[2]t^2+a[3]t^3+...+a[n]t^n=0
Is expressed as,
a[0]+a[1]ct+a[2]c^2t^2+a[3]c^3t^3+...+a[n]c^nt^n=0
Where c is almost 1 then dividing the two,
a[1]ct+a[2]c^2t^2+a[3]c^3t^3+...+a[n]c^nt^n= -a[0]
---------------------------------------------------
a[1]t+a[2]t^2+a[3]t^3+...+a[n]t^n = -a[0]
Suppose K=(a[1]c,a[2]c^2,a[3]c^3,...a[n]c^n) then simply,
T*(K-N)=0 so (K-N) is orthogonal to T. Consequently,
N*(K-N)
T is parallel to N - ------- K and
|K-N|^2
N*(K-N)
[m(N - -------- K) - Q]*N=0 solve this for m. Then
|K-N|^2
N*(K-N)
T= m(N - -------- K)
|K-N|^2
Substitute m in the above and take the square of the
magnitude of both sides. Then
(K-N)*(K-N) 0
|T|^2=Lim (-a[0])^2 ---------------------------- = ---
c->1 |N|^2(K-N)*(K-N)-[N*(K-N)]^2 0
applying LÕHopital two times with respct to c,
|C|^2
|T|^2=(-a[0])^2{-------------------}
|N|^2 |C|^2-(N*C)^2
where
d
C = Lim ---- K = =(a[1],2a[2],3a[3],4a[4],...,na[n])
c->1 dc
E.O.P.
Jon Giffen
http://mypeoplepc.com/members/jon8338/polynomial/id7.html
===
Subject: Re: Root Finder 12
> Root Finder 12
> by Jon Giffen.
> This solution was so simple that I couldnÕt believe it.
> But I tried it, and it works.
> It is found that the roots to the polynomial,
> a[0]+a[1]t+a[2]t^2+...+a[n]t^n=0 where
> T=(t,t^2,t^3,..,t^n) and
> N=(a[1],a[2],a[3],...,a[n]) are given by,
> -a[0]
> T= -----D
> D*N
> D=(1,2,3,4,...,n)
Ok then, attempt to construct the quadratic formula for your
method:
at^2 + bt + c = 0
T = (t, t^2)
N = (b, a)
D = (1, 2)
T = -c/(b+2a) * (1,2) = (t, t^2)
t = -c/(b+2a)
t^2 = -2c/(b+2a)
Clearly you are dead wrong, since the correct tÕs are
t = (-b + sqrt[b^2-4ac])/(2a)
and
t = (-b - sqrt[b^2-4ac])/(2a)
If you can not reconstruct the quadratic formula, then you
are wrong.
Any polynomials with a root approximated by your method are
just
coincidences.
===
Subject: Re: Root Finder 12
ex.
t^2+2t-3=0
N=(2,1) |N|^2=5 Q=(3/5)(2,1) D=(1,2) |D|^2=5 Q*D=12/5
(mD-Q)*D=0
m|D|^2=Q*D m=(Q*D)/|D|^2 = 12/25
T=mD = (12/25)(1,2)
t^2=24/25 ~ 1
>>Root Finder 12
>>by Jon Giffen.
>>This solution was so simple that I couldnÕt believe it.
>>But I tried it, and it works.
>>It is found that the roots to the polynomial,
>>a[0]+a[1]t+a[2]t^2+...+a[n]t^n=0 where
>>T=(t,t^2,t^3,..,t^n) and
>>N=(a[1],a[2],a[3],...,a[n]) are given by,
>> -a[0]
>>T= -----D
>> D*N
>>D=(1,2,3,4,...,n)
> Ok then, attempt to construct the quadratic formula for
your method:
> at^2 + bt + c = 0
> T = (t, t^2)
> N = (b, a)
> D = (1, 2)
> T = -c/(b+2a) * (1,2) = (t, t^2)
> t = -c/(b+2a)
> t^2 = -2c/(b+2a)
> Clearly you are dead wrong, since the correct tÕs are
> t = (-b + sqrt[b^2-4ac])/(2a)
> and
> t = (-b - sqrt[b^2-4ac])/(2a)
> If you can not reconstruct the quadratic formula, then you
are wrong.
> Any polynomials with a root approximated by your method are
just
> coincidences.
===
Subject: Re: Root Finder 12
> Root Finder 12
> by Jon Giffen.
> This solution was so simple that I couldnÕt believe it.
> But I tried it, and it works.
> [...]
> Solve the 6th degree polynomial,
> f(t)=t^6 + t - 10=0
> a[0] = -10 N=(1,0,0,0,0,1) D=(1,2,3,4,5,6)
> a[0]=-10 D*N=7
> -a[0] 10
> T = -----D = ---(1,2,3,4,5,6)
> D*N 7
> t =10/7 t=1.42835 f(1.42835)=-0.071
10/7 is not a root of t^6 + t - 10 = 0. The Rational Root
Test says
that any rational solutions to this polynomial are +/-1,
+/-2, +/5,
or +/10. It can only be an approximation.
Strike Twelve.
(The Rational Root Theorem -- a.k.a. the Rational Zero
Theorem --
can be found at
http://mathworld.wolfram.com/RationalZeroTheorem.html .
> Solve the 49th degree polynomial,
> f(t)=t^49 + t^16 + 40t^5 - 6000 = 0
> a[0]=-6000 N=(1,0,0,0,40,0,..,1,0,...,1) D=(1,2,3,..,49)
> D*N=1+200+16+49=266
> -a[0] 6000
> T = -----D = ------(1,2,3,..,49)
> D*N 266
> t^49=294000/266=1105.26 t=1.15375 f(1.15375)=-3575.99
Once again, the Rational Root Theorem says that the only
possible rational
roots are integers.
> Applying NewtonÕs Method,
Ah, so. You arenÕt \finding roots after all, \
only
approximations to them.
YouÕve been told repeatedly that this isnÕt \
the same as
\finding the roots.
>
t=1.15375-(-3575)/[49(1.15375^48)+16(1.15375^15)+200(1.15375)
^4]
> =1.15375 again, so the answer must depend on distant decimal
> places.
The problem here is you donÕt have enough precision to make
NewtonÕs
Method work.
> [...]
> t^2 + 2t - 3 = 0
> a[0]=-3 N=(2,1) D=(1,2) D*N=4
> -a[0] 3
> T = -----D =---(1,2) t^2=3/4 t=0.866 ~ 1
> D*N 4
What? Your method canÕt even solve a quadratic equation?
ThatÕs when
you know itÕs really bad.
-- Christopher Heckman
===
Subject: Re: Root Finder 12
ex.
t^2+2t-3=0
N=(2,1) |N|^2=5 Q=(3/5)(2,1) D=(1,2) |D|^2=5 Q*D=12/5
(mD-Q)*D=0
m|D|^2=Q*D m=(Q*D)/|D|^2 = 12/25
T=mD = (12/25)(1,2)
t^2=24/25 ~ 1
ex.
at^2+bt+c=0
N=(b,a) |N|^2=b^2+a^2 Q=(-c/[b^2+a^2])(b,a) D=(1,2) |D|^2=5
Q*D=(-c/[b^2+a^2])(b+2a)
(mD-Q)*D=0
m=(Q*D)/|D|^2 = (1/5)(-c/[b^2+a^2])(b+2a)
T=mD=(1/5)(-c/[b^2+a^2])(b+2a)(1,2)
t^2 =(2/5)(-c/[b^2+a^2])(b+2a)
b+/-{b^2-4ac}^(1/2)
t ={(2/5)(-c/[b^2+a^2])(b+2a)}^(1/2)=--------------------
2a
solve and \find the required correction
>>Root Finder 12
>>by Jon Giffen.
>>This solution was so simple that I couldnÕt believe it.
>>But I tried it, and it works.
>>[...]
>>Solve the 6th degree polynomial,
>>f(t)=t^6 + t - 10=0
>>a[0] = -10 N=(1,0,0,0,0,1) D=(1,2,3,4,5,6)
>>a[0]=-10 D*N=7
>> -a[0] 10
>>T = -----D = ---(1,2,3,4,5,6)
>> D*N 7
>>t =10/7 t=1.42835 f(1.42835)=-0.071
> 10/7 is not a root of t^6 + t - 10 = 0. The Rational Root
Test says
> that any rational solutions to this polynomial are +/-1,
+/-2, +/5,
> or +/10. It can only be an approximation.
> Strike Twelve.
> (The Rational Root Theorem -- a.k.a. the Rational Zero
Theorem --
> can be found at
http://mathworld.wolfram.com/RationalZeroTheorem.html .
>>Solve the 49th degree polynomial,
>>f(t)=t^49 + t^16 + 40t^5 - 6000 = 0
>>a[0]=-6000 N=(1,0,0,0,40,0,..,1,0,...,1) D=(1,2,3,..,49)
>>D*N=1+200+16+49=266
>> -a[0] 6000
>>T = -----D = ------(1,2,3,..,49)
>> D*N 266
>>t^49=294000/266=1105.26 t=1.15375 f(1.15375)=-3575.99
> Once again, the Rational Root Theorem says that the only
possible
rational
> roots are integers.
>>Applying NewtonÕs Method,
> Ah, so. You arenÕt \finding roots after all, \
only
approximations to them.
> YouÕve been told repeatedly that this isnÕt \
the same as
\finding the
roots.
>>t=1.15375-(-3575)/[49(1.15375^48)+16(1.15375^15)+200(
1.15375)^4]
>> =1.15375 again, so the answer must depend on distant
decimal
>>places.
> The problem here is you donÕt have enough precision to \
make
NewtonÕs
> Method work.
>>[...]
>>t^2 + 2t - 3 = 0
>>a[0]=-3 N=(2,1) D=(1,2) D*N=4
>> -a[0] 3
>>T = -----D =---(1,2) t^2=3/4 t=0.866 ~ 1
>> D*N 4
> What? Your method canÕt even solve a quadratic equation?
ThatÕs when
> you know itÕs really bad.
> -- Christopher Heckman
===
Subject: Re: Root Finder 12
I thought you already posted the last word on this subject.
Are you
suffering from some kind of attention de\ficit disorder, or are
you
being deliberately misleading? If thereÕs a third
possibility, IÕd
welcome your explanation.
--
There are two things you must never attempt to prove: the
unprovable
-- and the obvious.
--
Democracy: The triumph of popularity over principle.
--
http://www.crbond.com
===
Subject: Re: Root Finder 12
You people are savage, and thereÕs no call for it. I suppose
you are one of the anal perfectionist obsessed with keeping on
course down to the Angstrom to offset the Gudermanian.
NewtonÕs Method is an invention of genius. Why not use it?
Newton just didnÕt come up with the approximations to plug
into
it...
or did he?
> I thought you already posted the last word on this subject.
Are you
> suffering from some kind of attention de\ficit disorder, or
are you
> being deliberately misleading? If thereÕs a third
possibility, IÕd
> welcome your explanation.
> --
> There are two things you must never attempt to prove: the
unprovable
> -- and the obvious.
> --
> Democracy: The triumph of popularity over principle.
> --
> http://www.crbond.com
===
Subject: Re: Root Finder 12
ThereÕs always room for improvement
> I thought you already posted the last word on this subject.
Are you
> suffering from some kind of attention de\ficit disorder, or
are you
> being deliberately misleading? If thereÕs a third
possibility, IÕd
> welcome your explanation.
> --
> There are two things you must never attempt to prove: the
unprovable
> -- and the obvious.
> --
> Democracy: The triumph of popularity over principle.
> --
> http://www.crbond.com
===
Subject: Re: Root \finder XI
> Root Finder xi.
> by Jon Giffen.
> It is found that the roots to the polynomial,
> a[0]+a[1]t+a[2]t^2+...+a[n]t^n where
> [...]
> f(t) = t^7 + t^3 + t - 20 = 0
> [...]
> applying this to NewtonÕs Method,
YouÕve been told about this before. There are already good
places to
start off NewtonÕs Method, so you havenÕt \
provided anything
new.
Strike Eleven.
-- Christopher Heckman
===
Subject: Re: Root \finder XI
> Root Finder xi.
You told us that ix was the \final one.
===
Subject: Re: Root \finder XI
>
> Root Finder xi.
>
> You told us that ix was the \final one.
Based on his posts, did you really think you could believe
him?
-- Christopher Heckman
===
Subject: 20th Century obsession with self defeating statements
posting-account=Qiuj5AwAAACmGnmS12qcvqA9IXzD0s4L
Why are mathematicians focussed on the impossible?
What impossible entities have they proven?
A program that examines itself, if it halts then continues?
A number that is different to all other listed numbers, only
by an
explicit in\finite clause that concatinates the opposing digit
of all
listed in\finite numbers.
A statement that asserts you canÕt prove me?
A *\finite* sized algorithm that gives the maximum output
length of
*any* sized algorithm?
A set that doesnÕt contain itself, yet contains all sets \
that
donÕt
contain themselves?
Mathematical history is full of tragedy, read up on any great
mathematician and you will \find his life ended either early or
in
sorrow. Be objective with the proofs handed to you, most of
the above
are just a narrowly de\fined domain and a self referential
negative
clause de\fined on naive interpretations of that domain,
speci\fically
the de\fining clauses of the domain are just twisted.
Cardinal representation is a shorthand for numbers, a digit
by some
power of 10, the implied connotation is that digit is
selected from a
total set of digits, Cantors speci\fically de\fined \
opposing
digit is
obscure.
Herc
===
Subject: Re: Simple Group Theory Question
<co02k6$1m95$1@agate.berkeley.edu>
posting-account=jcZk7AwAAADXpPEyHtVyWC264SxtppRB
One more comment makes things
pretty easy. If you know about commutators, use
a^2 = 1 for all a to calculate the commutators, which gives
the thm Arturo Magidin speaks of.
Van
===
Subject: Daryl answer the question please.......
posting-account=Qiuj5AwAAACmGnmS12qcvqA9IXzD0s4L
How many digits d is D matched to on any new in\finite random
list?
IÕm not constructing a list given your number, you can put
your number
in a black box and hold the key.
THEN I generate a countable in\finite random list.
THEN you hand over the key.
THEN we check how many digits of your number I matched on my
INDPENDANT
list.
How many digits d is D matched to on any *new* in\finite
*random*
list?
Herc
>How many digits d is D matched to on any *new* in\finite
*random* list?
I donÕt know.
--
Josh Purinton
Does anyone know?
Is it
AA/ 0
A/ 1
B/ >1
C/ <10
D/ >10
E/ >1,000,000
F/ unlimited
G/ in\finite
H/ all of them
how many
digits a countable random list will match any given number?
Do any people who follow Cantors proof know how many digits
of your
NEW sequence will be matched?
===
Subject: Re: Daryl answer the question please.......
HERC777 says...
>How many digits d is D matched to on any new in\finite random
list?
>IÕm not constructing a list given your number, you can put
your number
>in a black box and hold the key.
>THEN I generate a countable in\finite random list.
>THEN you hand over the key.
>THEN we check how many digits of your number I matched on my
INDPENDANT
>list.
>How many digits d is D matched to on any *new* in\finite
*random*
>list?
Okay, assume that weÕre producing our digits by some truly
random
physical process (such as radioactive decay) so that each
digit is
equally likely and is statistically independent of all
previously
generated digits. Let R(i) be random real number i, where
R(0) is
my real. Then whatÕs likely to be the case is the following:
1. For every n > 0, there exists a number k(n) such that
R(0) agrees with R(k(n)) in the \first n decimal places.
2. For every k > 0, there exists a number n(k) such that
R(0) disagrees with R(k) at position n(k).
So there would be no bound on the number of decimal places
that match,
but none of your reals would be exactly equal to my real.
--
Daryl McCullough
Ithaca, NY
===
Subject: Re: Daryl answer the question please.......
<co0qua02cn4@drn.newsguy.com>
posting-account=Qiuj5AwAAACmGnmS12qcvqA9IXzD0s4L
1 -> for every digit place, there is a person from the
in\finite list
that matches up to that digit
2 -> for every person, there exists a digit on his sequence
that is
different
why does 2 hold? are you saying 2 random processes never have
the same
output? are you working backwards from whatÕs likely,
assuming the
in\finite set is incomplete?
Herc
===
Subject: Re: Daryl answer the question please.......
HERC777 says...
>1 -> for every digit place, there is a person from the
in\finite list
>that matches up to that digit
>2 -> for every person, there exists a digit on his sequence
that is
>different
>why does 2 hold? are you saying 2 random processes never
have the same
>output?
Right. With probability 1, no two random processes produce
the same
(in\finite) output. The probability that two randomly generated
sequences
of digits will agree on the \first n digits is 10^{-n}. So the
probability
that they will agree *everywhere* is the limit as n -->
in\finity
of 10^{-n}, which is 0.
>are you working backwards from whatÕs likely, assuming the
>in\finite set is incomplete?
Well, IÕm assuming that if you have countably many events,
each of
which is probability 0, then the probability of all of them
together
is still probability 0.
--
Daryl McCullough
Ithaca, NY
===
Subject: Re: Daryl answer the question please.......
<co10b002sd6@drn.newsguy.com>
posting-account=Qiuj5AwAAACmGnmS12qcvqA9IXzD0s4L
Why would you use a statistical method for \finite sets here?
With in\finite outcomes, each individual oucome is P=0.
P=0 means possible in this context.
If HHHHHHHHH.. is in the range of outputs, and in\finite trials
are
performed, there is no limit to the number of heads.
Logical conclusion, all heads.
Herc
===
Subject: Re: Daryl answer the question please.......
says...
>How many digits d is D matched to on any new in\finite random
list?
>IÕm not constructing a list given your number, you can put
your number
>in a black box and hold the key.
>THEN I generate a countable in\finite random list.
>THEN you hand over the key.
>THEN we check how many digits of your number I matched on my
INDPENDANT
>list.
>How many digits d is D matched to on any *new* in\finite
*random*
>list?
>Herc
>>How many digits d is D matched to on any *new* in\finite
*random* list?
>I donÕt know.
>Josh Purinton
>Does anyone know?
>Is it
>AA/ 0
>A/ 1
>B/ >1
>C/ <10
>D/ >10
>E/ >1,000,000
>F/ unlimited
>G/ in\finite
>H/ all of them
CLUE how many
>digits a countable random list will match any given number?
>Do any people who follow Cantors proof know how many digits
of your
>NEW sequence will be matched?
===
Subject: David Ullrich answer the questions please...........
posting-account=Qiuj5AwAAACmGnmS12qcvqA9IXzD0s4L
An in\finite number of people each toss a coin \
in\finite times.
Can you
guarantee a new sequence of heads and tails?
A ____
You hand the sequence to me in a locked box and keep the key.
Then I generate a second in\finite random list of H&T \
sequences.
Then you hand me the key.
How many digits of your sequence did I match the second time?
(the list is independant of the sequence)
A_____
Herc
===
Subject: Re: David Ullrich answer the questions
please...........
>An in\finite number of people each toss a coin \
in\finite times.
Can you
>guarantee a new sequence of heads and tails?
Well, since you said please:
I donÕt have any idea what youÕre asking. What \
do you mean,
a new sequence?
(My best guess is you mean a sequence thatÕs never been seen
before. But that makes no sense, because _no_ in\finite \
sequence
has ever been seen.)
>A ____
>You hand the sequence to me in a locked box and keep the key.
Ah, would that we could.
>Then I generate a second in\finite random list of H&T
sequences.
>Then you hand me the key.
>How many digits of your sequence did I match the second time?
>(the list is independant of the sequence)
If both sequences are generated randomly and the two sequences
are independent then of course itÕs impossible to say with
certainty how many matches there are - random is like that.
But with probability one there will be in\finitely many
matches, in fact the set of places where the two sequences
match will have asymptotic density 1/2.
>A_____
>Herc
************************
David C. Ullrich
===
Subject: Re: David Ullrich answer the questions
please...........
<4cs8q0tn63pf5p4p5g33e75ieqruat41s8@4ax.com>
posting-account=Qiuj5AwAAACmGnmS12qcvqA9IXzD0s4L
9 results for new number ullrich cantor
[Herc]
An in\finite number of people each toss a coin \
in\finite times
[DU]
because _no_ in\finite sequence has ever been seen.
20 people understood the question so far, do you want a
clause on the
second sentence saying within the domain of the 1st sentence?
Daryl
has this problem too, P=0 so an in\finite sequence is
impossible to
make... unless you use the diagonal.
IÕll rephrase in case you can answer.
When many many people all toss coins many many times, can you
toss your
coin in such a way that it makes a new sequence of heads and
tails that
no other person has made with their coin?
If there are n people, by the binomial distribution and with
90%
con\fidence interval, you will have to toss your coin approx.
log(n)
times.
What happens as n->oo?
IÕm amazed such a large group of people here in sci.math
donÕt recoil
from the obvious error in uncountable theory and see the
logically
obvious, that with in\finite people, the sequences all covered
are
in\finite in length.
Herc
you donÕt think that Klingon trick will really work do you?
===
Subject: Re: David Ullrich answer the questions
please...........
>9 results for new number ullrich cantor
>[Herc]
>An in\finite number of people each toss a coin \
in\finite times
>[DU]
>because _no_ in\finite sequence has ever been seen.
>20 people understood the question so far, do you want a
clause on the
>second sentence saying within the domain of the 1st
sentence? Daryl
>has this problem too, P=0 so an in\finite sequence is
impossible to
>make... unless you use the diagonal.
>IÕll rephrase in case you can answer.
>When many many people all toss coins many many times, can
you toss your
>coin in such a way that it makes a new sequence of heads and
tails that
>no other person has made with their coin?
ThatÕs a repharasing of An in\finite number of \
people each
toss a coin in\finite times. Can you
guarantee a new sequence of heads and tails? ? many many
is a rephrasing of in\finite? Also in the \first \
question
you asked whether I could guarantee a sequence with a
certain property, now you ask whether I _can_ toss it in
a certain way...
The answer to the second version of the question depends
on (i) how many people toss a coin, and how many times
(ii) whether youÕre asking (a) can I be certain that
my sequence will be different or (b) is it possible
that my sequence is different, if IÕm allowed to control
my sequence.
>If there are n people, by the binomial distribution and with
90%
>con\fidence interval, you will have to toss your coin approx.
log(n)
>times.
>What happens as n->oo?
>IÕm amazed such a large group of people here in sci.math
donÕt recoil
>from the obvious error in uncountable theory and see the
logically
>obvious, that with in\finite people, the sequences all covered
are
>in\finite in length.
Yeah, that _is_ hard to understand...
>Herc
>you donÕt think that Klingon trick will really work do you?
************************
David C. Ullrich
===
Subject: Re: David Ullrich answer the questions
please...........
<4cs8q0tn63pf5p4p5g33e75ieqruat41s8@4ax.com>
<3d1aq0hcipbkk03i9scg3u6vfoprm94co8@4ax.com>
posting-account=Qiuj5AwAAACmGnmS12qcvqA9IXzD0s4L
OK, IÕm going to see if you can answer this new question,
because I
fully know you are mistaken that you can \find a new \
(different)
sequence when that sequence is fully within_the_range of an
effectively
identical person in a set of in\finite people all trying to
copy you.
Try to get the gist of it and work out my intended meaning if
there is
ambiguity.
Given
1/ an in\finite sequence D
2/ a set S of s in\finite lists of in\finite \
sequences
Given s is suf\ficiently large, what portion of S contain a
\finite
maximum to the initial length of D that is matched?
Say D = <HHHHHH..>
Say S1 = {
<HTHTHTHT..>
<TTTTHTHTH..>
<HHHTHTHT..>
...
}
Assume S1 due to rare random ßuctuations does not contain
in\finite H..
Then it has some \finite limit, it could be 3 by the example
above.
Basically, what is the con\fidence interval that an \
in\finite
list does
not contain some given sequence?
Will it tend to always match it to in\finite digits?
Will it tend to contain some \finite limit?
99% of the time, any given sequence [WILL] / [WILL NOT] be
matched to
in\finite precision on a random in\finite list.
Herc
===
Subject: Re: David Ullrich answer the questions
please...........
>OK, IÕm going to see if you can answer this new question,
because I
>fully know you are mistaken that you can \find a new
(different)
>sequence when that sequence is fully within_the_range of an
effectively
>identical person in a set of in\finite people all trying to
copy you.
Beezarre. You keep changing the question - now the others are
trying to copy me? _Previously_ they went \first. You really
need
to make up your mind what the question is...
>Try to get the gist of it and work out my intended meaning
if there is
>ambiguity.
>Given
>1/ an in\finite sequence D
>2/ a set S of s in\finite lists of in\finite \
sequences
Is S countable?
>Given s is suf\ficiently large, what portion of S contain a
\finite
>maximum to the initial length of D that is matched?
This question makes no sense unless I assume that s was a
typo for
S.
Assuming that, the question makes no sense. What initial
length
are you talking about? What the heck is a \finite maximum to
an initial length?
Oh. Maybe you mean to ask what portion of S contains an
initial segment of maximal length matching D.
ItÕs obviously impossible to answer this question without
be that none of the sequences in S match D at _all_. Or it
could be that all the sequences in S are exactly the same
as D.
>Say D = <HHHHHH..>Say S1 = {
><HTHTHTHT..><TTTTHTHTH..><HHHTHTHT..>...
>Assume S1 due to rare random ßuctuations does not contain
in\finite H..
>Then it has some \finite limit,
Huh? I have no idea what it means to say S1 has some \finite
limit.
(Hint: this is because it makes no sense to say that.)
>it could be 3 by the example above.
>Basically, what is the con\fidence interval that an \
in\finite
list does
>not contain some given sequence?
>Will it tend to always match it to in\finite digits?
>Will it tend to contain some \finite limit?
>99% of the time, any given sequence [WILL] / [WILL NOT] be
matched to
>in\finite precision on a random in\finite list.
>Herc
************************
David C. Ullrich
===
Subject: Re: David Ullrich answer the questions
please...........
<4cs8q0tn63pf5p4p5g33e75ieqruat41s8@4ax.com>
<3d1aq0hcipbkk03i9scg3u6vfoprm94co8@4ax.com>
<m7ibq0dl5dg3q7656thp2faht3seo4llgp@4ax.com>
posting-account=Qiuj5AwAAACmGnmS12qcvqA9IXzD0s4L
>OK, IÕm going to see if you can answer this new question,
because I
>fully know you are mistaken that you can \find a new
(different)
>sequence when that sequence is fully within_the_range of an
effectively
>identical person in a set of in\finite people all trying to
copy you.
Beezarre. You keep changing the question - now the others are
trying to copy me? _Previously_ they went \first. You really
need
to make up your mind what the question is...
[Herc]
[read the OP, the 1st question is you try to be different to
the
in\finite set,
the 2nd question is another in\finite set tries to copy you]
>Try to get the gist of it and work out my intended meaning
if there is
>ambiguity.
>Given
>1/ an in\finite sequence D
>2/ a set S of s in\finite lists of in\finite \
sequences
Is S countable?
[Herc]
[Yes, S is a \finite set. s e N. S = SET s = size]
>Given s is suf\ficiently large, what portion of S contain a
\finite
>maximum to the initial length of D that is matched?
This question makes no sense unless I assume that s was a
typo for
S.
[Herc]
[Given the size (s) of the set (S) is large enough to get an
expected
random distrution]
Assuming that, the question makes no sense. What initial
length
are you talking about? What the heck is a \finite maximum to
an initial length?
Oh. Maybe you mean to ask what portion of S contains an
initial segment of maximal length matching D.
[Herc - yes]
ItÕs obviously impossible to answer this question without
be that none of the sequences in S match D at _all_. Or it
could be that all the sequences in S are exactly the same
as D.
[Herc]
[ThatÕs why S is actually a set of S1, S2, S3... Ss
Assuming s is large enough, what is the _typical_ behaviour
for any
Sx?]
>Say D = <HHHHHH..>Say S1 = {
><HTHTHTHT..><TTTTHTHTH..><HHHTHTHT..>...
>Assume S1 due to rare random ßuctuations does not contain
in\finite
H..
>Then it has some \finite limit,
Huh? I have no idea what it means to say S1 has some \finite
limit.
(Hint: this is because it makes no sense to say that.)
[Herc]
[S1 contains an initial segment of maximum length matching D.
I meant
bound not limit. Note that S1 may contain an initial segment
of maximum
length matching D, and S2 may contain D]
>it could be 3 by the example above.
>Basically, what is the con\fidence interval that an \
in\finite
list does
>not contain some given sequence?
>Will it tend to always match it to in\finite digits?
>Will it tend to contain some \finite limit?
>99% of the time, any given sequence [WILL] / [WILL NOT] be
matched to
>in\finite precision on a random in\finite list.
>Herc
************************
David C. Ullrich
[Herc]
===
Subject: Re: please help me
>>A baseball is popped straight up with an initial velocity
of 32
>>feet per second,
>Giving KE = 1/2 m v^2 = 1/2 m * (32fps)^2
>>What is the maximum height
>>reached by this baseball?
>It is the height at which PE=KE, and PE+KE=k throughout the
trajectory.
No, it is the height at which velocity is 0, thereby forcing
KE to be
0 also.
>PE = mgh = m*32fpsps*h
>1/2 m * (32 ft / sec)^2 = m * (32 ft/sec^2) * h
>m factors out
>1/2 * 1024 ft^2/sec^2 = 32 ft/sec^2 * h(ft)
>512 ft^2/sec^2 / 32 ft/sec^2 = h
>16 ft = h.
>ItÕs an energy balance.
>Now apply KE=PE throughout the trajectory to determine
>> its height above the ground is a function of
>>time
>I tolerance everything and tolerate everyone.
>I love: Dona, Jeff, Kim, Kimmie, Mom, Neelix, Tasha, and
Teri,
alphabetically.
>I drive: A double-step Thunderbolt with 657% range.
>I \fight terrorism by: Using less gasoline.
<<Remove the del for email>>
===
Subject: ? solving linear eqn
Given A*x = b, here x and b are vectors and A is a matrix,
square or not.
I saw the following:
1) given A, b and then solve for b
but how about the following:
2) given x and b, solve for A?
by Cheng Cosine
Nov/23/2k4 Ut
===
Subject: Re: ? solving linear eqn
> Given A*x = b, here x and b are vectors and A is a matrix,
square or not.
> I saw the following:
> 1) given A, b and then solve for b
> but how about the following:
> 2) given x and b, solve for A?
Someone suggested to extend (2) into more general form that
all A, x, and b are matrix. Then I can easily analysis A*x = b
as what textbook usually taught to do for A*x = b as usual.
But this approach just remind me another linear equation
below.
A*x+x*B = C
Here all are matrices. A is m-by-m and B is n-by-n, while both
x and C are m-by-n.
I read in some linear system books called this as Lyapunov
equation,
and some interesting theorems are given. However, I donÕt \
see
how
to analysize linear problems of this kind. To be more
speci\fic, in
analyzing
A*x = b, I read ppl use the eigenvector or spectral expansion
or more
generally the SVD. Then one can see when the solution exist
and unique.
But I donÕt see any way to do analysis for A*x+x*B = C.
Any suggestions?
by Cheng Cosine
Nov/24/2k4 UT
===
Subject: Re: ? solving linear eqn
> Given A*x = b, here x and b are vectors and A is a matrix,
square or
not.
> I saw the following:
> 1) given A, b and then solve for b
> but how about the following:
> 2) given x and b, solve for A?
> Someone suggested to extend (2) into more general form that
> all A, x, and b are matrix. Then I can easily analysis A*x
= b
> as what textbook usually taught to do for A*x = b as usual.
> But this approach just remind me another linear equation
below.
> A*x+x*B = C
> Here all are matrices. A is m-by-m and B is n-by-n, while
both
> x and C are m-by-n.
> I read in some linear system books called this as Lyapunov
equation,
> and some interesting theorems are given. However, I donÕt
see how
> to analysize linear problems of this kind. To be more
speci\fic, in
analyzing
> A*x = b, I read ppl use the eigenvector or spectral
expansion or more
> generally the SVD. Then one can see when the solution exist
and unique.
> But I donÕt see any way to do analysis for A*x+x*B = C.
> Any suggestions?
> by Cheng Cosine
> Nov/24/2k4 UT
Cheng, that someone is quite right. Let us \first consider how
to solve
an equation in square matrices
(1) A * X = B,
With A and B given, provided A is of maximum rank, in other
words
det(A) is unequal 0 (but for ChristÕs sake never bother to
compute a
determinant, use the methods suggested below instead), the
solution X
will then be found by \finding the inverse of A, call it \
inv(A),
multiply both sides with inv(A) on the left and we get, since
inv(A) *
A = E (the unity matrix with the diagonal elements =1, all
other
elements =0), E*X=inv(A)*B, and since of course the unity
matrix
multiplied with any matrix leaves that matrix unchanged, we
\find
X=inv(A)*B. What your original question started from, albeit
with a
matrix A but a vector b and an unknown vector x, was to turn
things
aroung and ask for A, given x and b. Guess what, in terms of
matrices
the new viewpoint is not so drastically different from (1),
we merely
now talk about a matrix equation
(2) X * A = B,
and again, provided A [ has maximum rank/has a non-zero
determinant/is
invertable] - you guessed it, the three conditions stated in
the
square brackets are equivalent, provided the condition is
met, the
solution is obtained by forming inv(A), multiplying both
sides (2)
with inv(A) but on the right this time, and get X=B*inv(A).
The main computational task here obviously is to \find the
inverse of
A. Using the time-honoured methods \first developed by
C.F.Gauss you
adjoin the unity matrix E to the right of A, call that
structure (A|E)
and through the repeated application of elementary row
operations
bring the A part of that adjoined structure to diagonal form.
Call
that transformed diagonal matrix t(A), and the simultaneously
transformed unity matrix t(E). If t(A) is not maximum rank,
ie. it has
0 elements in the diagonal, i.e. its determinant is 0,
rejoice since
(1), or (2), have no solution and your work is done. If
however t(A)
has maximum rank you now have a structure (t(A)|t(E)) where
t(A) is a
diagonal matrix and t(E) certainly no longer looks like a
unity
matrix. By taking the column vectors of t(E) one by one, call
them c
and solving for vectors x in
(3) t(A) * x = c
you have obtained inv(A) as the successive x vectors are none
other
than the columns of inv(A). For 3x3 or 4x4 matrices the
entire Ôs
computation can be done on a notepad and need not take longer
than 60
seconds. With a bit of practice your lecturer will accuse you
cheating!
At long last to your particular problem. Given vectors a and
b solve
for a matrix X so that
(4) X * a = b
Simple. Transform a to a matrix A by interpreting a as the
\first
column vector of A, keep adding columns until you have a
square
matrix, A. A will need to have an inverse, so you have to keep
checking. For 2- or 3- vectors this is trivial, but for larger
structures I am sure one could \find an algorithm of the
successive
application of elementary row operations a la C.F.Gauss so
you donÕt
just randomly form A from a to then \find that the result does
not have
maximum rank. Let us now assume you have adjoined a to a
suitable A.
You may now adjoining any columns at all to b to transform
this vector
into a square matrix B and BINGO, the solution will be:
(5) X = B * inv(A)
as per the exercise we did for (2).
So, ZVK has given you the professional mathematicianÕs \
answer
to your
problem, I have provided the market gardenerÕs version, what
more
could you ask for?
Actually, a few very important things require an answer in
connection
with (4), and I leave these for you as an exercise:
(a) are there vectors a and b for which an X cannot be found?
SIMPLE,
just look at (5) and ask yourself what a would make it
impossible for
A to have an inverse;
(b) obviously if there is one solution X there is an in\finity
of them,
and it would certainly be desirable to express this in\finite
solution
space in terms of a space, ie. show its dimension and perhaps
write
down elements that allow the general solution to be described
as a
linear combination of these elements.
Good luck. Reinhard
PS.
===
Subject: Re: ? solving linear eqn
> Given A*x = b, here x and b are vectors and A is a matrix,
square or not.
> I saw the following:
> 1) given A, b and then solve for b
> but how about the following:
> 2) given x and b, solve for A?
> by Cheng Cosine
> Nov/23/2k4 Ut
In 1), you probably meant ... then solve for x.
Question 2), provided x is not the zero vector, may have
in\finitely many
solutions.
Just play around with
A being 1-by-2 unknown [u, v],
x a 2-by-1 column vector with 1Õs as entries,
b a 1-by-1 matrix [1].)
The question was somewhat playful anyway, but
without any effort to see whatÕs to be expected.
Want to see more? (If not, skip the rest. You will need
something like
normed linear algebra to understand it.)
Serious treatment comes under the name backward error
analysis:
Suppose one has obtained an approximate solution x of the
equation
C * x = b with given C and b.
A question can be asked
is x an exact solution of a system A*x=b where A is close to
C?
Write A=C+H, then we are looking for a small H so that
(C+H)*x=b. That means H*x=b-C*x (b-C*x is called the
residual).
If the norms of matrices and vectors are compatible
(submultiplicative),
then norm(H) * norm(x) >= norm(b-C*x), so that
norm(H) >= norm(b-C*x) / norm(x).
Can the lower bound be achieved? Actually yes, and I will
give the
construction for Euclidean norm:
set H = (b-C*x)*x^transposed / norm(x)^2.
Your question, to summarize, has an answer A = C + H.
For other norms, one would have to play with the dual norm,
and a
\finite-dimensional version of Hahn-Banach Theorem.
Sorry you asked? :-)
===
Subject: Re: Automorphisms of abelian p-groups:
Aut(Z_p+Z_{p^2})
>> What is the automorphism group of Z_p+Z_{p^2} for p prime?
> This is what I have so far: Each automorphism must \fixes the
class of
>> elements of order p. So, it must \fixes the subgroup
Z_p+Z_p. But IÕm
>> not sure how to continue. Can someone please lead me a
little?
>G = C_p x C_{p^2}, with presentation <x,y | x^{p^2}, y^p>,
clearly
>has p^2*(p-1) elements of order p^2 -- each cyclic subgroup
of
>order p^2 has p(p-1) such elements, and there are p such
cyclic
>subgroups: <x>, <xy>, ..., <xy^{p-1}> -- and p(p-1) elements
of
>order p that arenÕt powers of elements of order p^2 -- the
>non-identity elements in <y>, <x^p y>, ..., <x^{p(p-1)} y>
-- so
>|Aut(G)| = p^3*(p-1)^2.
>automorphisms for Aut(G) = <A,B,C,D>, with A^{p(p-1)} = 1,
>B^p = 1, C^{p-1} = 1, D^p = 1:
>A(x) = x^n, A(y) = y, where n generates (Z_{p^2},*) =~
C_{p(p-1)}
>B(x) = xy, B(y) = y
>C(x) = x, C(y) = y^m, where m generates (Z_p,*) =~ C_{p-1}
>D(x) = x, D(y) = x^p y
>Filling in the details of the presentation of Aut(G), i.e.,
>\finding suf\ficient relations among the \
generators, is left as
an
>exercise for the reader.
> You might have added that Aut(G) has order p^3 (p-1)^2,
See the last clause of my \first, admittedly not very clearly
written, paragraph.
> with a normal
> extraspecial group of order p^3 (generated by A^(p-1), B,
D, with A^(p-1)
> central), and the quotient group is isomorphic to Z_{p-1} x
Z_{p-1}.
above, rather than discovering it later while playing around
with relations among the generators of Aut(G). (Why do you
think
I left that as an exercise for the reader? :-)
--
Jim Heckman
===
Subject: Re: Determination of sine frequency
>does anybody know a convenient method for the determination
of the
frequency
>of a sine wave when exactly four aquidistant samples are
known (A0, A1,
A2,
>A3). The tricky point is that these four samples do not
cover one complete
>period (this would be easy determined by the discrete
fourier transform)
but
>only one part of one period.
If f(x) = k * sin(x*2*pi*f),
fÕÕ(x) = -k * (2*pi*f)^2 * sin(x*2*pi*f)
You can estimate fÕÕ(A1) = (f(A2)-f(A0))/t^2
Where t is the interval between samples.
This assumes that the data is normalized to the range [-1..1]
and probably
some other things I didnÕt consider.
--Keith Lewis klewis {at} mitre.org
The above may not (yet) represent the opinions of my employer.
===
Subject: Quantum Gravity Hypothesis
The physicist Lee Smolin explains that space is relational,
very
analogous to being inside the words of a sentence: The
geometry of
a universe is very like the grammatical structure of a
sentence. Just
as a sentence has no structure and no existence apart from the
relationships between the words, space has no existence apart
from the
relationships that hold between the things in the universe.
Coordinates are a convienient means of mathematical modeling,
...but
reality actually de\fines itself with events. Events \
donÕt
happen in
a \fixed absolute background of space, or time; events
characterize the
evolution OF space-time. The metric of space-time becomes
de\fined
by events, such, that there is no space-time if there are no
events.
A metric \field can be de\fined by the primary \
substratum of
events.
Thus the intrinsic geometrical structure of spacetime is
predicated on
the pseudo-Riemannian spaces via the af\fine relationships all
physical events are fully reducible to manifestations of the
substratum i. e. the event density generating a metric \field.
The overlap of events as ripples- being the wave functions -
circular
conic 2D cross sections, generates new smaller ripples that
are
contained in the outer past ripples-cross sections, thus the
locality principle is not violated and describes
non-paradoxically,
what appears as non-local transfer of information.
[event_2^0_[[[_[_[event 2^n_]___]_]]]]]
Intersections[the overlapping] of event boundaries also
provides a
better de\finition for bits of physical information, where the
information density of the universe is continually increasing.
Time is de\fined as an iterative sequence of outer[past] events
including all inner[future] events.
http://www.iomas.com/gina/ultrahiq/Mega-Society/NoesisMay/
SupernovaCL.asp
http://www.martinelli.org/rexpansion/
===
Subject: Re: Quantum Gravity Hypothesis
Russell E. Rierson <analog57@yahoo.com> schrieb
> The physicist Lee Smolin explains that space is relational,
very
> analogous to being inside the words of a sentence: The
geometry of
> a universe is very like the grammatical structure of a
sentence. Just
> as a sentence has no structure and no existence apart from
the
> relationships between the words, space has no existence
apart from the
> relationships that hold between the things in the universe.
Nice example. The point is that all real sentences we hear or
read have
some other structure. For example, this sentence has also the
structure
of a stream of bits. Acoustic sentences have the structure of
sound
waves.
Ilja
===
Subject: Re: Quantum Gravity Hypothesis
> The physicist Lee Smolin explains that space is relational,
very
> analogous to being inside the words of a sentence: The
geometry of
> a universe is very like the grammatical structure of a
sentence. Just
> as a sentence has no structure and no existence apart from
the
> relationships between the words, space has no existence
apart from the
> relationships that hold between the things in the universe.
http://www.edge.org/3rd_culture/smolin/smolin_p3.html
We see that, at least naively time has completely disappeared
from
the formalism. This has led to what is called the problem of
time in
quantum cosmology, which is how to [] provide an
interpretation
according to which time is not part of a fundamental
description of
the world, but only reappears in an appropriate classical
limit.
If he progresses to substituting expression for description he
will have turned an important corner, not just regarding the
concept
time but also observer.
Of course, this too is the essential problem with space-time.
It
connotes a clumsy (in a complex world), primal concept of
observer -
which is consistent with the general primal understanding of
the
nature of Nature as empirical.
http://www.effectuationism.com/forum/messages/27/27.html?
1071620499
With further development: Inde\finite and dynamic Man/Person-
-Ground
in tension with moving animal/object - Time- -Being.
Peter Kinane
http://www.effectuationism.com/
===
Subject: Re: Quantum Gravity Hypothesis
> The physicist Lee Smolin explains that space is relational,
very
> analogous to being inside the words of a sentence: The
geometry of
> a universe is very like the grammatical structure of a
sentence. Just
> as a sentence has no structure and no existence apart from
the
> relationships between the words, space has no existence
apart from the
> relationships that hold between the things in the universe.
...[trim]...
Also take the time to check CahillÕs work in this \
\field:
www.mountainman.com.au/process_physics
The geometry of the universe might be generated
out of randomness.
===
Subject: Re: Quantum Gravity Hypothesis
> The physicist Lee Smolin explains that space is relational,
very
> analogous to being inside the words of a sentence: The
geometry of
> a universe is very like the grammatical structure of a
sentence. Just
> as a sentence has no structure and no existence apart from
the
> relationships between the words, space has no existence
apart from the
> relationships that hold between the things in the universe.
That is just another way of stating that we use mathematical
MODELS of
the world. Inherently such models are not real.
> Coordinates are a convienient means of mathematical
modeling,
Hmmm. Say manifolds rather than coordinates, because thatÕs
what modern
physical theories actually use for this modeling.
> ...but
> reality actually de\fines itself with events. Events \
donÕt
happen in
> a \fixed absolute background of space, or time; events
characterize the
> evolution OF space-time. The metric of space-time becomes
de\fined
> by events,
Sure. See above.
> such, that there is no space-time if there are no events.
But in any interesting physical situation there are always
events.
HereÕs an example list:
Object A exists at proper time t0
Object A exists at proper time t1
Object A exists at proper time t2
...
Object B ...
Clearly for a classical theory in which proper time is
continuous the
events are not countable as this particular enumeration
suggests.
> [... excursion into never-never land]
Tom Roberts tjroberts@lucent.com
===
Subject: Re: Quantum Gravity Hypothesis
> The physicist Lee Smolin explains that space is relational,
very
> analogous to being inside the words of a sentence: The
geometry of
> a universe is very like the grammatical structure of a
sentence. Just
> as a sentence has no structure and no existence apart from
the
> relationships between the words, space has no existence
apart from the
> relationships that hold between the things in the universe.
> Coordinates are a convienient means of mathematical
modeling, ...but
> reality actually de\fines itself with events. Events \
donÕt
happen in
> a \fixed absolute background of space, or time; events
characterize the
> evolution OF space-time. The metric of space-time becomes
de\fined
> by events, such, that there is no space-time if there are
no events.
> A metric \field can be de\fined by the primary \
substratum of
events.
> Thus the intrinsic geometrical structure of spacetime is
predicated on
> the pseudo-Riemannian spaces via the af\fine relationships ?
all
> physical events are fully reducible to manifestations of the
> substratum i. e. the event density generating a metric \
\field.
I think the wording is sloppy. An event is a point
in 4 coordinates x,y,z,t. The term event density
*might* refer to spacetime \field density, but do I
have to guess?
> The overlap of events as ripples- being the wave functions
- circular
> conic 2D cross sections, generates new smaller ripples that
are
> contained in the outer past ripples-cross sections,
These so-called ripples imply an occurance.
An occurance is much more complex than an event.
An occurance needs dx,dy,dz,dt,de (e=energy) at
some fuzzy location x,y,z,t.
The de, usually a photon, proves the existance
of matter, which in turn, from the PoV of GR tells
spacetime where it is. Hence we survey spacetime
using photons, as a result of the occurances.
Bilge and I had a discussion about this before,
and Bilge pointed out that we should not screw
with the de\finition of what an event is, leave
it as x,y,z,t and I agree.
>thus the
> locality principle is not violated and describes
non-paradoxically,
> what appears as non-local transfer of information.
> [event_2^0_[[[_[_[event 2^n_]___]_]]]]]
> Intersections[the overlapping] of event boundaries
You see, the term event boundaries is poor
vocabulary, (hey I used it and was rightly corrected),
and should be termed occurance boundaries,
because dx,dy,dz,dt,de is fuzzy.
Occurances continually over-lap.
>also provides a
> better de\finition for bits of physical information, where \
the
> information density of the universe is continually
increasing.
A non-zero divergence of information. No wonder
the universe will always be smarter than us!
> Time is de\fined as an iterative sequence of outer[past]
events
> including all inner[future] events.
I think the ISU has de\fined physical time very
well, if one intends to introduce new de\finitions
of time, they must be able to transform that de\finition
to the ISU standard or justify a change in that standard.
Did Lee Smolin really say this in all seriousness?
Ken S. Tucker
===
Subject: Re: Quantum Gravity Hypothesis
> The physicist Lee Smolin explains that space is relational,
very
> analogous to being inside the words of a sentence: The
geometry
of
> a universe is very like the grammatical structure of a
sentence. Just
> as a sentence has no structure and no existence apart from
the
> relationships between the words, space has no existence
apart from the
> relationships that hold between the things in the universe.
>
> Coordinates are a convienient means of mathematical
modeling, ...but
> reality actually de\fines itself with events. Events \
donÕt
happen in
> a \fixed absolute background of space, or time; events
characterize the
> evolution OF space-time. The metric of space-time becomes
de\fined
> by events, such, that there is no space-time if there are
no events.
>
> A metric \field can be de\fined by the primary \
substratum of
events.
> Thus the intrinsic geometrical structure of spacetime is
predicated on
> the pseudo-Riemannian spaces via the af\fine relationships ?
all
> physical events are fully reducible to manifestations of the
> substratum i. e. the event density generating a metric \
\field.
> I think the wording is sloppy. An event is a point
> in 4 coordinates x,y,z,t. The term event density
> *might* refer to spacetime \field density, but do I
> have to guess?
The book Gravitation - by Misner, Thorne, and Wheeler[1st
chapter],
says that points in spacetime are characterized by what
happens
there. Give a point in spacetime the name event. World lines
of
Worldlines \fill up spacetime. Events relate to other events.
> The overlap of events as ripples- being the wave functions
- circular
> conic 2D cross sections, generates new smaller ripples that
are
> contained in the outer past ripples-cross sections,
> These so-called ripples imply an occurance.
> An occurance is much more complex than an event.
> An occurance needs dx,dy,dz,dt,de (e=energy) at
> some fuzzy location x,y,z,t.
> The de, usually a photon, proves the existance
> of matter, which in turn, from the PoV of GR tells
> spacetime where it is. Hence we survey spacetime
> using photons, as a result of the occurances.
Yes, an event is a happening or an occurance.
http://dictionary.reference.com/search?q=event&r=67
QUOTE:
event
Something that takes place; an occurrence.
A signi\ficant occurrence or happening. See Synonyms at
occurrence.
A social gathering or activity.
The \final result; the outcome.
Sports. A contest or an item in a sports program.
Physics. A phenomenon or occurrence located at a single point
in
space-time, regarded as the fundamental observational entity
in
relativity theory.
END OF QUOTE
> Bilge and I had a discussion about this before,
> and Bilge pointed out that we should not screw
> with the de\finition of what an event is, leave
> it as x,y,z,t and I agree.
Tensors are coordinate independent. In the real world,
events[occurances] relate to other events[occurances]. The
real world
doesnÕt require a superimposing of Cartesian, or other
coordinates, on
it.
>thus the
> locality principle is not violated and describes
non-paradoxically,
> what appears as non-local transfer of information.
>
> [event_2^0_[[[_[_[event 2^n_]___]_]]]]]
>
> Intersections[the overlapping] of event boundaries
> You see, the term event boundaries is poor
> vocabulary, (hey I used it and was rightly corrected),
> and should be termed occurance boundaries,
> because dx,dy,dz,dt,de is fuzzy.
The terms Event and occurance are interchangable for this
purpose.
> Occurances continually over-lap.
>also provides a
> better de\finition for bits of physical information, where \
the
> information density of the universe is continually
increasing.
> A non-zero divergence of information. No wonder
> the universe will always be smarter than us!
> Time is de\fined as an iterative sequence of outer[past]
events
> including all inner[future] events.
> I think the ISU has de\fined physical time very
> well, if one intends to introduce new de\finitions
> of time, they must be able to transform that de\finition
> to the ISU standard or justify a change in that standard.
The above de\finition doesnÕt contradict the ISU \
standards.
> Did Lee Smolin really say this in all seriousness?
> Ken S. Tucker
Yes.
http://books.guardian.co.uk/reviews/scienceandnature/0%
2C6121%2C438833%2C00.
html
QUOTE
But in EinsteinÕs theory, space is something else \
altogether.
It
arises out of the relationships between objects - planets,
galaxies
and so on. Think of a sentence, says Smolin; it is not simply
a
container into which one puts words. Without any words, there
would be
no sentence; similarly, in EinsteinÕs theory, space has no
existence
apart from the objects that move within it.
END OF QUOTE
===
Subject: Re: Quantum Gravity Hypothesis
...
Russell, I think you did good post.
> A metric \field can be de\fined by the primary \
substratum of
events.
> Thus the intrinsic geometrical structure of spacetime is
predicated
on
> the pseudo-Riemannian spaces via the af\fine relationships ?
all
> physical events are fully reducible to manifestations of the
> substratum i. e. the event density generating a metric \
\field.
>
> I think the wording is sloppy. An event is a point
> in 4 coordinates x,y,z,t. The term event density
> *might* refer to spacetime \field density, but do I
> have to guess?
> The book Gravitation - by Misner, Thorne, and Wheeler[1st
chapter],
> says that points in spacetime are characterized by what
happens
> there. Give a point in spacetime the name event. World
lines of
> Worldlines \fill up spacetime. Events relate to other events.
Yes, the universe has many photons following worldlines
for reference, but when you actually measure one, by
aborption or emission, Heisenburg makes us limit our
accuracy to dx,dy,dz,dt,de, which I term an occurance.
In basic relativity we *simplify* the occurance to be
an event at x,y,z,t, but then we leave the real world
of measurement and go into an imaginary world of continuous
manifolds independent of energy and matter, that is
inconsistent with GR and QT.
> The overlap of events as ripples- being the wave functions
- circular
> conic 2D cross sections, generates new smaller ripples that
are
> contained in the outer past ripples-cross sections,
>
> These so-called ripples imply an occurance.
> An occurance is much more complex than an event.
> An occurance needs dx,dy,dz,dt,de (e=energy) at
> some fuzzy location x,y,z,t.
> The de, usually a photon, proves the existance
> of matter, which in turn, from the PoV of GR tells
> spacetime where it is. Hence we survey spacetime
> using photons, as a result of the occurances.
> Yes, an event is a happening or an occurance.
> http://dictionary.reference.com/search?q=event&r=67
> QUOTE:
> event
>
> Something that takes place; an occurrence.
> A signi\ficant occurrence or happening. See Synonyms at
occurrence.
> A social gathering or activity.
> The \final result; the outcome.
> Sports. A contest or an item in a sports program.
> Physics. A phenomenon or occurrence located at a single
point in
> space-time, regarded as the fundamental observational
entity in
> relativity theory.
Close but if you want to invite Webster to lecture on
physics, well you know what I mean,
> END OF QUOTE
> Bilge and I had a discussion about this before,
> and Bilge pointed out that we should not screw
> with the de\finition of what an event is, leave
> it as x,y,z,t and I agree.
> Tensors are coordinate independent. In the real world,
> events[occurances] relate to other events[occurances]. The
real world
> doesnÕt require a superimposing of Cartesian, or other
coordinates, on
> it.
Basically thatÕs true. But a tensor must be able to
be specialized as a set of measurements wrt some
arbituary CS. So somewhere in the process one will
need to be *able* to de\fine an event at x,y,z,t, by
an occurance, dx...de, at the point of measurement.
>thus the
> locality principle is not violated and describes
non-paradoxically,
> what appears as non-local transfer of information.
>
> [event_2^0_[[[_[_[event 2^n_]___]_]]]]]
>
> Intersections[the overlapping] of event boundaries
>
> You see, the term event boundaries is poor
> vocabulary, (hey I used it and was rightly corrected),
> and should be termed occurance boundaries,
> because dx,dy,dz,dt,de is fuzzy.
> The terms Event and occurance are interchangable for this
purpose.
IÕm not sure. ItÕs my impression Smolin is \
presenting
an analogy for a description of quantizing GR, not an
easy thing to do, understandably. But I object to
*bastardizing* (no offense intended), the de\finition
of event.
His ripples in spacetime are better descibed as the
result of occurances because they involve energy.
> Occurances continually over-lap.
>
>also provides a
> better de\finition for bits of physical information, where \
the
> information density of the universe is continually
increasing.
>
> A non-zero divergence of information. No wonder
> the universe will always be smarter than us!
>
> Time is de\fined as an iterative sequence of outer[past]
events
> including all inner[future] events.
>
> I think the ISU has de\fined physical time very
> well, if one intends to introduce new de\finitions
> of time, they must be able to transform that de\finition
> to the ISU standard or justify a change in that standard.
> The above de\finition doesnÕt contradict the \
ISU standards.
OK, but it seems like a complicated way to describe a Cesium
clock, unless he has something different in mind.
> Did Lee Smolin really say this in all seriousness?
> Ken S. Tucker
> Yes.
>
http://books.guardian.co.uk/reviews/scienceandnature/0%
2C6121%2C438833%2C00.h
tml
> QUOTE
> But in EinsteinÕs theory, space is something else
altogether. It
> arises out of the relationships between objects - planets,
galaxies
> and so on. Think of a sentence, says Smolin; it is not
simply a
> container into which one puts words. Without any words,
there would be
> no sentence; similarly, in EinsteinÕs theory, space has no
existence
> apart from the objects that move within it.
> END OF QUOTE
Ken S. Tucker
===
Subject: Re: Quantum Gravity Hypothesis
> The physicist Lee Smolin explains that space is relational,
very
> analogous to being inside the words of a sentence: The
geometry of
> a universe is very like the grammatical structure of a
sentence. Just
> as a sentence has no structure and no existence apart from
the
> relationships between the words, space has no existence
apart from the
> relationships that hold between the things in the universe.
> Coordinates are a convienient means of mathematical
modeling, ...but
> reality actually de\fines itself with events. Events \
donÕt
happen in
> a \fixed absolute background of space, or time; events
characterize the
> evolution OF space-time. The metric of space-time becomes
de\fined
> by events, such, that there is no space-time if there are
no events.
Snip to make my point.
But we know there is empty space-time because we can move
into it
Russell. We also know that empty space-time curves because
of gravity. The metric is independent of any events there.
The patterns of curved space-time around mass are absolute.
Albert
Einstein
Einstein asked not about what atoms are; he said; but instead
he
wanted to know what was inbetween them.
You may question what empty space-time means but not its
existence.
Mitch Raemsch -- Light Falls --
===
Subject: Re: Quantum Gravity Hypothesis
> The physicist Lee Smolin explains that space is relational,
very
> analogous to being inside the words of a sentence: The
geometry of
> a universe is very like the grammatical structure of a
sentence. Just
> as a sentence has no structure and no existence apart from
the
> relationships between the words, space has no existence
apart from the
> relationships that hold between the things in the universe.
> Coordinates are a convienient means of mathematical
modeling, ...but
> reality actually de\fines itself with events. Events \
donÕt
happen in
> a \fixed absolute background of space, or time; events
characterize the
> evolution OF space-time. The metric of space-time becomes
de\fined
> by events, such, that there is no space-time if there are
no events.
> A metric \field can be de\fined by the primary \
substratum of
events.
> Thus the intrinsic geometrical structure of spacetime is
predicated on
> the pseudo-Riemannian spaces via the af\fine relationships -
all
> physical events are fully reducible to manifestations of the
> substratum i. e. the event density generating a metric \
\field.
> The overlap of events as ripples- being the wave functions
- circular
> conic 2D cross sections, generates new smaller ripples that
are
> contained in the outer past ripples-cross sections, thus the
> locality principle is not violated and describes
non-paradoxically,
> what appears as non-local transfer of information.
> [event_2^0_[[[_[_[event 2^n_]___]_]]]]]
> Intersections[the overlapping] of event boundaries also
provides a
> better de\finition for bits of physical information, where \
the
> information density of the universe is continually
increasing.
> Time is de\fined as an iterative sequence of outer[past]
events
> including all inner[future] events.
Reading the above I am reminded of the cartoon where you see
a physicist,
systems analyst, mathematician , engineer or whatever with
all these
arcane
symbols leading to a box with a sign that says Ôand then a
miracle occursÕ.
Someone comes along and looks at it and says - good work but
you might want
to tighten your reasoning up here - pointing to the bit that
says Ôand then
a miracle occursÕ. For some reason when I read the above I \
am
reminded of
the cartoon and feel an irresistible urge to say - you might
like to
tighten
you reasoning up here. The only difference is at least you
understand
what - Ôand then a miracle occurs- means. \
AFAICS the above
is basically
unintelligible gibberish. But then again what the heck would
I know - the
physics books I read usually express their ideas in
mathematical language
not words whose meaning is as obscure as the book of
revelations.
Bill
>
http://www.iomas.com/gina/ultrahiq/Mega-Society/NoesisMay/
SupernovaCL.asp
> http://www.martinelli.org/rexpansion/
===
Subject: get me a printout on that Data!
posting-account=Qiuj5AwAAACmGnmS12qcvqA9IXzD0s4L
[Spock]
Captain the Klingons are attempting to hijack our computer.
[Kirk]
Scotty! Full power dammit!
[Spock]
Captain, theyÕve locked their quantum drive computer into \
our
in\finite
encryptable login mainframe.
[Kirk]
Those pesky Klingons, if their quantum computer tried EVERY
COMBINATION
in parallel weÕll be goners!
[Spock]
DonÕt worry captain, IÕve \
identi\fied their quantum device, its
an
Enter-prizonator 3000, IÕm diagonalising now...
[Kirk]
Great Spock! You should be a lecturer!
[Spock]
IÕve reprogrammed our computer security Captain, the \
in\finite
login
sequence is now set to the inverted diagonal of the
Enterprizonator
3000, its spockandthepointyearedchick...
[Kirk]
DonÕt tell me, its a secret remember, good work crew, warp
factor 9!
Herc
===
Subject: Re: get me a printout on that Data!
> [Spock]
> Captain the Klingons are attempting to hijack our computer.
> [Kirk]
> Scotty! Full power dammit!
> [Spock]
> Captain, theyÕve locked their quantum drive computer into
our in\finite
> encryptable login mainframe.
> [Kirk]
> Those pesky Klingons, if their quantum computer tried EVERY
COMBINATION
> in parallel weÕll be goners!
> [Spock]
> DonÕt worry captain, IÕve \
identi\fied their quantum device,
its an
> Enter-prizonator 3000, IÕm diagonalising now...
> [Kirk]
> Great Spock! You should be a lecturer!
> [Spock]
> IÕve reprogrammed our computer security Captain, the
in\finite login
> sequence is now set to the inverted diagonal of the
Enterprizonator
> 3000, its spockandthepointyearedchick...
> [Kirk]
> DonÕt tell me, its a secret remember, good work crew, warp
factor 9!
> Herc
Herc your stuff is getting better!
===
Subject: Re: get me a printout on that Data!
posting-account=Qiuj5AwAAACmGnmS12qcvqA9IXzD0s4L
Herc
===
Subject: US usage of the terms college and university
Can some US reader enlighten me as to the usage of the terms
college
and university in your country? Does, for example, a college
math
course have the same conotation as a university math course?
They
are quite different here in Canada.
Dan
Toronto, Canada
===
Subject: Re: US usage of the terms college and university
>Can some US reader enlighten me as to the usage of the terms
college
>and university in your country? Does, for example, a college
math
>course have the same conotation as a university math course?
As others have noted, there is no standard distinction. (As
they say,
The nice thing about standards is that there are so many to
choose
from!)
To add another usage: the departments of my university are
organized into
seven colleges: the College of Liberal Arts and Sciences, the
College of Education, the College of Law, etc.
You might \find this useful:
http://www.math.niu.edu/~rusin/teaching-math/usa
for non-USAns of the US school system.
I have heard surprised foreign visitors compare our College
Algebra
course to College Alphabet. Sad to say, at least a quarter of
US
post-secondary students need to take this course.
dave
===
Subject: Re: US usage of the terms college and university
>>Can some US reader enlighten me as to the usage of the
terms college
>>and university in your country? Does, for example, a
college math
>>course have the same conotation as a university math course?
> As others have noted, there is no standard distinction. (As
they say,
> The nice thing about standards is that there are so many to
choose
from!)
> To add another usage: the departments of my university are
organized into
> seven colleges: the College of Liberal Arts and Sciences,
the
> College of Education, the College of Law, etc.
> You might \find this useful:
> http://www.math.niu.edu/~rusin/teaching-math/usa
> for non-USAns of the US school system.
> I have heard surprised foreign visitors compare our College
Algebra
> course to College Alphabet. Sad to say, at least a quarter
of US
> post-secondary students need to take this course.
> dave
ItÕs sad. Heck, it was a shock to me. I spent my
undergraduate career at
a school where the lowest level math class available was a
calculus
class intended for people who never took HS calc. So when I
went to grad
school and saw many sections of a class called Algebra and
Trigonometry I was rather taken aback.
-Ron
===
Subject: Re: US usage of the terms college and university
3QLpj-NoP*NzsIC,boYU]bQ]HÕ<P%JD/,.J>y<#4ga3$21:
> Can some US reader enlighten me as to the usage of the
terms college
> and university in your country? Does, for example, a
college math
> course have the same conotation as a university math
course? They
> are quite different here in Canada.
My understanding is that in most cases the difference is that
a college
has only undergraduate programs, a university has both
undergraduate and
graduate programs. But it is also not uncommon for a
university to have
subdivisions called colleges, representing either academic
units (UCI
College of Medicine) or housing units (Kresge College at
UCSC).
Anyway, I wouldnÕt expect much difference between the two
math course
phrases you describe.
--
David Eppstein
Computer Science Dept., Univ. of California, Irvine
http://www.ics.uci.edu/~eppstein/
===
Subject: Re: US usage of the terms college and university
>> Can some US reader enlighten me as to the usage of the
terms college
>> and university in your country? Does, for example, a
college math
>> course have the same conotation as a university math
course? They
>> are quite different here in Canada.
>My understanding is that in most cases the difference is
that a college
>has only undergraduate programs, a university has both
undergraduate and
>graduate programs. But it is also not uncommon for a
university to have
>subdivisions called colleges, representing either academic
units (UCI
>College of Medicine) or housing units (Kresge College at
UCSC).
>Anyway, I wouldnÕt expect much difference between the two
math course
>phrases you describe.
WhatÕs the difference between Harvard College and Harvard
University?
Harvard College is the undergraduate program at Harvard. It
is part of
the Faculty of Arts and Sciences and offers programs in the
liberal
arts.
Harvard University refers to the entire educational
institution,
including the undergraduate college, the graduate and
professional
schools, research centers, administration, and af\filiates.
===
Subject: Re: US usage of the terms college and university
> Can some US reader enlighten me as to the usage of the
terms college
> and university in your country?
There is no standardized distinction between college and
university in the United States. Many Americans think that
there is,
but there is not.
> Does, for example, a college math
> course have the same conotation as a university math course?
The connotation of college math course is math course
encountered
in higher education (that is, tertiary education) rather than
what an
American would call a high school math course, that is math
course
encountered in secondary education. Of course, some people in
the
United States repeat in higher education courses that they
ought to
have learned in secondary education. Moreover, much of the
precalculus
mathematics that some Americans donÕt take until
college/university is
actually a junior-high subject in many other parts of the
world, e.g.,
Taiwan.
> They are quite different here in Canada.
ItÕs a puzzlement to people from many countries that
Americans donÕt
draw a sharp terminological distinction between colleges and
universities, but it is, alas, a fact.
Hope this helps! When I write to international audiences,
such as this
newsgroup, I usually try to remember to say university when
referring to my alma mater (which was indeed denominated a
university)
just so that I donÕt confuse readers from other countries.
But when
speaking to other Americans, I might simply say, e.g., When I
went to
college . . . to refer to that same institution of higher
education.
--
Karl M. Bunday P.O. Box 1456, Minnetonka MN 55345
Learn in Freedom (TM) http://learninfreedom.org/
remove .de to email
===
Subject: Re: US usage of the terms college and university
A college can be a junior college, also called a community
college; or it
may
be a university. In The USA, a university is a college but a
college is
not
always a university.
A university is a college. A university permits the study to
earn an
bachelor
of arts degree, complete some certi\ficate program, or earn
master of arts
or
master of science degree. Some universities have programs for
earning PhD.
A
university is a 4 year institution, the frequent amount of
time expected
for
earning bachelor of arts or bachelor of science degrees.
A university is a college, and offers a higher range of study
than
community
colleges (which are also called junior colleges).
A community college or junior college permits study to earn a
lower degree,
called associate in arts degree; also for studying in some
certi\ficate
programs. A community college is a 2 year institution. The
A.A. degree
can
be earned in about two years.
Be aware, sometimes a student will use more than 2 years of
study at the
community college. Students change their major \field of study,
sometimes
enroll
in fewer courses during a semester; maybe take time off to do
other things.
The same variations happen by students at universities. Some
students take
more than 4 years at a university to earn a Bachelor degree.
In summary, distinction between college an university is more
adequately
characterised as so:
Community College or Junior College:
2 year college; AA degree, some certi\ficate programs, courses
for personal
interest, vocational training programs.
University( also a college):
4 year college; BA & BS Degrees, Masters degree, at some,
PhDs. Some
certi\ficate programs
G C
===
Subject: Re: US usage of the terms college and university
> Can some US reader enlighten me as to the usage of the
terms college
> and university in your country? Does, for example, a
college math
> course have the same conotation as a university math
course? They
> are quite different here in Canada.
Both refer to tertiary, not secondary, institutions.
A college has only undergraduate, a university both
undergraduate and graduate, programs.
--
Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email)
===
Subject: Re: US usage of the terms college and university
>> Can some US reader enlighten me as to the usage of the
terms college
>> and university in your country? Does, for example, a
college math
>> course have the same conotation as a university math
course? They
>> are quite different here in Canada.
>Both refer to tertiary, not secondary, institutions.
>A college has only undergraduate, a university both
>undergraduate and graduate, programs.
This may or may not be the case. When Michigan State
College was upgraded to Michigan State University in
title, this was the only change. It was already a
university in all but name.
And there are of\ficially named universities which
do not give graduate courses, or give only a few
professional courses, which cannot be used for
graduate degrees.
--
This address is for information only. I do not claim that
these views
are those of the Statistics Department or of Purdue
University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
===
Subject: Re: US usage of the terms college and university
>Both refer to tertiary, not secondary, institutions.
>A college has only undergraduate, a university both
>undergraduate and graduate, programs.
> This may or may not be the case. When Michigan State
> College was upgraded to Michigan State University in
> title, this was the only change. It was already a
> university in all but name.
The sequence was Michigan State College of Agriculture and
Applied Science
(1925) -> Michigan State University of Agriculture and
Applied Science
(1955) -> Michigan State University (1964).
===
Subject: Re: US usage of the terms college and university
There was some Japanese movie a few years ago. The subtitles
said
a certain person would be starting college ... but it was
really more
like what we call middle school in the US. I donÕt know if
there is
a Japanese term college, or if the subtitle writer was from
some third
country where college has this meaning.
--
G. A. Edgar
http://www.math.ohio-state.edu/~edgar/
===
Subject: Re: US usage of the terms college and university
> Can some US reader enlighten me as to the usage of the terms
college
> and university in your country? Does, for example, a
college math
> course have the same conotation as a university math
course? They
> are quite different here in Canada.
> Both refer to tertiary, not secondary, institutions.
> A college has only undergraduate, a university both
> undergraduate and graduate, programs.
But TertiaryÕs what I did at 15-17 before I went to Uni.
Phil
--
... one Marine noticed one of the prisoners was still
breathing. A Marine
can be heard saying on the pool footage provided to Reuters
Television:
HeÕs fucking faking heÕs dead. He faking \
heÕs fucking dead.
The Marine then raises his riße and \fires into the \
manÕs head.
The pictures are too graphic for us to broadcast, Sites said.
===
Subject: Re: US usage of the terms college and university
> Can some US reader enlighten me as to the usage of the terms
college
> and university in your country? Does, for example, a
college math
> course have the same conotation as a university math
course? They
> are quite different here in Canada.
> Both refer to tertiary, not secondary, institutions.
> A college has only undergraduate, a university both
> undergraduate and graduate, programs.
An interesting opinion...
I have seen good colleges (complete with graduate
departments) become
universities simply by ordering new stationery.
===
Subject: Re: US usage of the terms college and university
>> Can some US reader enlighten me as to the usage of the
terms
college
>> and university in your country? Does, for example, a
college
math
>> course have the same conotation as a university math
course?
They
>> are quite different here in Canada.
>> Both refer to tertiary, not secondary, institutions.
>> A college has only undergraduate, a university both
>> undergraduate and graduate, programs.
>An interesting opinion...
>I have seen good colleges (complete with graduate
departments) become
>universities simply by ordering new stationery.
And I have seen (many) bad-to-middling colleges become
universities
simply by creating new (bad-to-bad) graduate programs (for
instance,
in hospitality). And, of course, ordering new stationery and
signage.
Lee Rudolph
===
Subject: Witzzle Pro Mail In Math Contest
posting-account=OS1-1g0AAAB-a0XMOi5PejhnEWKMANse
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awareness, learning and success for all children! Check back
in
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===
Subject: Re: solid angle
>Now that I come to think of it over my evening meal, there
was indeed
>a way of seeing this without doing the integrals. The area
of the unit
>sphere outside the cone is equal to that of a cylinder of
identical
>height and radius (=1). Now the height simply is equal to
the 2 *
>sinus(arctan(A/D).
>Frank
ThatÕs true of course, but to show an area on a sphere = the
area of
its projection on the cylinder is itself a calculus
(integration)
problem, no?
--Lynn
===
Subject: Re: solid angle
> ThatÕs true of course, but to show an area on a sphere =
the area of
> its projection on the cylinder is itself a calculus
(integration)
> problem, no?
> --Lynn
Yes, I believe that is true, although I have a notion that
Archimedes
had \figured this out already - but I do not know whether that
is
correct. Anyway, I only wanted to indicate that one could
understand
the equation using this knowledge. But I remain very happy
indeed that
you have explained the analytic underpinnings of these
connections.
Frank
===
Subject: Re: solid angle
>> ThatÕs true of course, but to show an area on a sphere =
the area of
>> its projection on the cylinder is itself a calculus
(integration)
>> problem, no?
>> --Lynn
>Yes, I believe that is true, although I have a notion that
Archimedes
>had \figured this out already - but I do not know whether that
is
>correct.
Archimedes did \figure it out, and by using the ideas of
integration, which have nothing to do with differential
calculus. Archimedes and Euclid and the Greeks educated
in Euclidean geometry understood limits and what we
call the Riemann integral, although they could not
often calculate it.
Consider the surface area of the frustrum of a cone which
comes close to matching a small slice of the sphere by
planes perpendicular to the cylinder tangent to the sphere.
Make sure the slope of the cone is close to that of the
that of the sphere at points of the sector. Then the
ratio of the area of the frustrum to that cut off on the
cylinder is close to one. The rest follows using limits.
One might say that this proof is not correct by modern
standards, and this can be argued, but not very well.
--
This address is for information only. I do not claim that
these views
are those of the Statistics Department or of Purdue
University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
===
Subject: Re: solid angle
>>Now that I come to think of it over my evening meal, there
was indeed
>>a way of seeing this without doing the integrals. The area
of the unit
>>sphere outside the cone is equal to that of a cylinder of
identical
>>height and radius (=1). Now the height simply is equal to
the 2 *
>>sinus(arctan(A/D).
>>Frank
>ThatÕs true of course, but to show an area on a sphere = \
the
area of
>its projection on the cylinder is itself a calculus
(integration)
>problem, no?
>--Lynn
No, Now that itÕs clear you are looking for the solid angle
of a
sphere with its polar caps lopped off, you can use PappusÕ
theorem
which says to get the area, multiply the arc length by 2piR,
the
equatorial path, so you get
a = atan(A/D)
arc = 2aR
Area = 2piR*2aR (and with R = 1),
= 4 pi atan(A/D) steradians
If you include sinus you get the cylindrical area equivalent,
not
reducible to steradians.
If you have something to say, write an equation.
If you have nothing to say, write an essay
===
Subject: Re: solid angle
>>ThatÕs true of course, but to show an area on a sphere =
the area of
>>its projection on the cylinder is itself a calculus
(integration)
>>problem, no?
>>--Lynn
>No, Now that itÕs clear you are looking for the solid angle
of a
>sphere with its polar caps lopped off, you can use PappusÕ
theorem
>which says to get the area, multiply the arc length by 2piR,
the
>equatorial path, so you get
> a = atan(A/D)
> arc = 2aR
> Area = 2piR*2aR (and with R = 1),
> = 4 pi atan(A/D) steradians
>
>If you include sinus you get the cylindrical area
equivalent, not
>reducible to steradians.
>If you have something to say, write an equation.
>If you have nothing to say, write an essay
IÕm not sure whether your response is directed at Frank or
myself, but
I will stick by my statement that to show an area on a sphere
= the
area of its projection on the cylinder is itself a calculus
(integration) problem. Even though the case of this
particular area
can be calculated via Pappustheorem, I would guess, not
having
looked at the proof recently, that you are just substituting
one
calculus result with another by using Pappustheorem. And
who would
know whether the original poster knew Pappustheorem?
--Lynn
===
Subject: Re: solid angle
>ThatÕs true of course, but to show an area on a sphere = \
the
area of
>its projection on the cylinder is itself a calculus
(integration)
>problem, no?
>--Lynn
>>No, Now that itÕs clear you are looking for the solid \
angle
of a
>>sphere with its polar caps lopped off, you can use PappusÕ
theorem
>>which says to get the area, multiply the arc length by
2piR, the
>>equatorial path, so you get
>> a = atan(A/D)
>> arc = 2aR
>> Area = 2piR*2aR (and with R = 1),
>> = 4 pi atan(A/D) steradians
>>
>>If you include sinus you get the cylindrical area
equivalent, not
>>reducible to steradians.
>>If you have something to say, write an equation.
>>If you have nothing to say, write an essay
>IÕm not sure whether your response is directed at Frank or
myself, but
>I will stick by my statement that to show an area on a
sphere = the
>area of its projection on the cylinder is itself a calculus
>(integration) problem. Even though the case of this
particular area
>can be calculated via Pappustheorem, I would guess, not
having
>looked at the proof recently, that you are just substituting
one
>calculus result with another by using Pappustheorem. And
who would
>know whether the original poster knew Pappustheorem?
>--Lynn
I goofed. I need to retract the Pappusschema; it needs the
centroid
of the arc to traverse the path. So I (cough, cough)
integrated
Int(2piR^2cos a da) = sin atan(A/D)*2pi *2 (R = 1)
But, ding, ding, sin(atan) can be simpli\fied so we get
Angle = 4pi/sqrt(1 + D^2/A^2)
This seems more appealing.
John Polasek
If you have something to say, write an equation.
If you have nothing to say, write an essay
===
Subject: quanti\fier conundrum
posting-account=oUD7iA0AAADl8hVsrZzNtstD0kITOUvN
i am reading d. vellemanÕs Ôhow to prove \
itand have come to
an
an exercise from the book below will illustrate my point.
where && is conjunction, || is disjunction, ! is not, E is the
existential quanti\fier and A is the universal \
quanti\fier.
the book gives:
let T(x,y) mean x teaches y
ExEy [ T(x,y) && !EuEv( T(u,v) && ( u != x || v != y ) ) ]
ExEy [ T(x,y) && AuAv!( T(u,v) && ( u != x || v != y ) ) ]
ExEy [ T(x,y) && AuAv( !T(u,v) || !( u != x || v != y ) ) ]
ExEy [ T(x,y) && AuAv( !T(u,v) || ( u = x && v = y ) ) ]
and several transformations later which i hope are correct,
we have:
ExEy [ T(x,y) && AuAv( T(u,v) -> ( u = x && v = y ) ) ]
which iÕd read as:
there exist one or more xÕs and one or more \
yÕs such that x
teaches y
and for all people u and all people v, if u teaches v then u
is x and
v is y.
is that the right way to read this statement?
anyway the main problem iÕm having is that iÕm \
still quite
murky on
exactly what x and y represent in this statement, because the
existential quanti\fier guarantees that there be one or more
things in
existence, so where it says u = x && v = y, which x and y is
this
referring to? in other words there may exist several xÕs and
yÕs where
x teaches y, so if for all u and v if u teaches v does that
mean that u
= the several x, and v the several y?
===
Subject: Re: quanti\fier conundrum
|ExEy [ T(x,y) && AuAv( T(u,v) -> ( u = x && v = y ) ) ]
[...]
|anyway the main problem iÕm having is that \
iÕm still quite
murky on
|exactly what x and y represent in this statement, because the
|existential quanti\fier guarantees that there be one or more
things in
|existence, so where it says u = x && v = y, which x and y is
this
|referring to?
Read the formula from the inside out. The subformula
T(x,y) && AuAv( T(u,v) -> ( u = x && v = y ) ) has two free
variables in it, x and y, so it describes a property of two
given
individuals x and y. Namely, it says that x teaches y, and
that
x is the only teacher and y is the only student (to paraphrase
a little). In relation to this subformula, x and y both refer
simply
to some given individuals. To get a de\finite statement one
would
have to supply the two individuals.
Now apply the existential quanti\fier Ey. You get a formula \
with
just one free variable, x, which says something about x.
Namely,
it says that x has (one or more) student(s) who have the
relationship
described in the previous paragraph to x. But of course there
can
be only one such student y. So it still means that x is the
only
teacher, and x has only one student.
Now apply the existential quanti\fier Ex. The resulting
sentence is
closed, i.e. no free variables, and says that one or more
people
have the property described in the previous paragraph. But of
course
there can be only one such teacher x. So it says thereÕs a
unique
teacher, and that the one teacher has only one student.
Keith Ramsay
===
Subject: Re: quanti\fier conundrum
> where && is conjunction, || is disjunction, ! is not, E is
the
> existential quanti\fier and A is the universal \
quanti\fier.
> let T(x,y) mean x teaches y
> ExEy [ T(x,y) && !EuEv( T(u,v) && ( u != x || v != y ) ) ]
> ExEy [ T(x,y) && AuAv!( T(u,v) && ( u != x || v != y ) ) ]
> ExEy [ T(x,y) && AuAv( !T(u,v) || !( u != x || v != y ) ) ]
> ExEy [ T(x,y) && AuAv( !T(u,v) || ( u = x && v = y ) ) ]
> and several transformations later which i hope are correct,
we have:
> ExEy [ T(x,y) && AuAv( T(u,v) -> ( u = x && v = y ) ) ]
Nicely done.
> which iÕd read as:
ThereÕs a x and a y such that T(x,y) and for all u,v
if T(u,v) then u = x, v = y
ThereÕs a x and a y such that T(x,y) and for all u,v
for which T(u,v), u = x, v = y.
ThereÕs a x and a y such that T(x,y) and any other u,v
with T(u,v) are x and y.
ThereÕs a x and a y such that T(x,y) and x and y are the \
only
such pair.
There is a unique x and y such that T(x,y)
> there exist one or more xÕs and one or more \
yÕs such that x
teaches y
> and for all people u and all people v, if u teaches v then
u is x and
> v is y.
> is that the right way to read this statement?
Very clumsy. The expression thereÕs one or more \
isnÕt
preferred except
perhaps for lax minds. Better and clearer is simply there is
a. In
contrast, thereÕs exactly one. There is one however may have
some
ambiguity about it.
> anyway the main problem iÕm having is that \
iÕm still quite
murky on
> exactly what x and y represent in this statement, because
the
> existential quanti\fier guarantees that there be one or more
things in
> existence, so where it says u = x && v = y, which x and y
is this
> referring to? in other words there may exist several xÕs
and yÕs where
> x teaches y, so if for all u and v if u teaches v does that
mean that u
> = the several x, and v the several y?
x and y are elements in the range of the quanti\fiers.
The statement says there is a x and a y such that T(x,y) and
...
The and ... part says that if thereÕs a u and a v with \
T(u,v),
then theyÕre x and y.
That one or more stuff I consider over complex, distracting
from,
rather than adding to clarity. I suggest you think
(Ex) there is a x and
(E!x) there is exactly one x or there is a unique x
and avoid thinking (Ex) there is one x as mildly ambiguous and
(Ex) there is or are, one or more than one, x or xÕs to give
itÕs
full and rigorous grammatical due, as abstrusely absurd. ;-)
Do you have a dollar? Yes I have a dollar.
Do you have \five? No, only two.
There you see, a already means one or more.
===
Subject: help! vector differentiation!
Hi all,
Can anybody help me on how to differentiate function of
vectors with respect
to a vector?
For example,
I know d(wÕ*A*w)/dw=2wÕ*A
where w is a column vector, A is symmetrical matrix, wÕ
denotes the
transpose.
wÕ*A*w is a scalar function of vector w.
The derivative of wÕ*A*w w.r.t. w = 2w* A
The second derivative of wÕ*A*w w.r.t. w = 2 A
-----------------------------------
But what should be the \first and second derivative of
(wÕ*A*w)*wÕ*B?
The reason I ask this because I want to \find second derivative
for
(wÕ*B*w) / (wÕ*A*w)
===
Subject: Re: help! vector differentiation!
> Hi all,
> Can anybody help me on how to differentiate function of
vectors with
respect
> to a vector?
> For example,
> I know d(wÕ*A*w)/dw=2wÕ*A
> where w is a column vector, A is symmetrical matrix, wÕ
denotes the
> transpose.
> wÕ*A*w is a scalar function of vector w.
> The derivative of wÕ*A*w w.r.t. w = 2w* A
> The second derivative of wÕ*A*w w.r.t. w = 2 A
> -----------------------------------
> But what should be the \first and second derivative of
(wÕ*A*w)*wÕ*B?
> The reason I ask this because I want to \find second
derivative for
> (wÕ*B*w) / (wÕ*A*w)
This result and many others are easily derived by using Ricci
notation, instead of the cumbersome matrix notation.
This particular result is in a short note on my website:
http://www.numerical-algorithms.com/
in the PDF dpocument:
http://www.numerical-algorithms.com/notes/ortho.pdf
equation (58).
In the usual matrix notation and a \fixed font this works out
to:
xT.A.x a
F = ------ = -
xT.B.x b
Fkl =
a [ 2 2 2 2 T T 2 2 T ]
- [ - A - - B - - - [(Ax)(Bx)+(Bx)(Ax)] + 2 - - (Bx)(Bx) ]
b [ a b a b b b ]
T
a = x.A.x
T
b = x.B.x
This by the way is a case of a matrix (A-lambda*B) perturbed
by an
outer product of vectors whose inverse can be expressed in
terms
of the inverse of (A-B).
Jentje Goslinga
===
Subject: Re: help! vector differentiation!
>> Hi all,
>> Can anybody help me on how to differentiate function of
vectors with
>> respect to a vector?
>> For example,
>> I know d(wÕ*A*w)/dw=2wÕ*A
>> where w is a column vector, A is symmetrical matrix, wÕ
denotes the
>> transpose.
>> wÕ*A*w is a scalar function of vector w.
>> The derivative of wÕ*A*w w.r.t. w = 2w* \
A
>> The second derivative of wÕ*A*w w.r.t. w = 2 A
>> -----------------------------------
>> But what should be the \first and second derivative of
(wÕ*A*w)*wÕ*B?
>> The reason I ask this because I want to \find second
derivative for
>> (wÕ*B*w) / (wÕ*A*w)
> This result and many others are easily derived by using
Ricci
> notation, instead of the cumbersome matrix notation.
> This particular result is in a short note on my website:
> http://www.numerical-algorithms.com/
> in the PDF dpocument:
> http://www.numerical-algorithms.com/notes/ortho.pdf
> equation (58).
> In the usual matrix notation and a \fixed font this works out
to:
> xT.A.x a
> F = ------ = -
> xT.B.x b
> Fkl =
> a [ 2 2 2 2 T T 2 2 T ]
> - [ - A - - B - - - [(Ax)(Bx)+(Bx)(Ax)] + 2 - - (Bx)(Bx) ]
> b [ a b a b b b ]
> T
> a = x.A.x
> T
> b = x.B.x
> This by the way is a case of a matrix (A-lambda*B)
perturbed by an
> outer product of vectors whose inverse can be expressed in
terms
> of the inverse of (A-B).
> Jentje Goslinga
Hi Jentje,
ThatÕs interesting... IÕd like to learn Ricci \
notation if it
outweighs the
cubersom matrix-vector notation...
But the paper on your website is not openable by my
Acroreader 6.0, what is
the problem?
===
Subject: Re: help! vector differentiation!
> Hi Jentje,
> ThatÕs interesting... IÕd like to learn \
Ricci notation if it
outweighs > the cubersom matrix-vector notation...
Well, there isnÕt much to learn about it, try calculating \
the
simple result you wanted yourself starting with:
Q = (akl xk xl)/(bmn xm xn) = a/b (no free indices)
Qi = d/dxi Q
Qi = [ail xl + aki xk]/b - (a/b^2) [bmi xm + bin xn]
= [ail xl + aik xk]/b - (a/b^2) [bim xm + bin xn]
= 2 (a/b) [(aik xk)/a - (bik xk)/b]
(because of the symmetry of A and B)
Qij = d/dxj Qi
Qij = 2 (a/b) [aij/a - bij/b - (aik xk)/a^2 (2 ajk xk)... ] +
2 [(aik xk)/a - (bik xk)/b] d/dxj (a/b)
Qij = 2 (a/b) [aij/a - bij/b - (aik xk)/a^2 (2 ajk xk)... ] +
2 [(aik xk)/a - (bik xk)/b] *
2(a/b)[(ajk xk)/a - (bjk xk)/b] (subst from previous)
and so on.
You can use the Indicial Tensor package in Maxima or Macsyma
for matrices without too many problems. Some of the results
in my little note on orthogonal matrices I managed to derive
by machine in this fashion.
Sorry, the result I gave you is for symmetric matrices which
I think is more or less implied by your question, at least
the denominator matrix (usually called B) in the generalized
eigenvalue problem is normally assumed to be symmetric and
positive de\finite, some kind of a norm.
For non-symmetric matrices you just retain aik xk and aki xk
separately throughout the calculation and translate them back
as Ax and ATx.
> But the paper on your website is not openable by my
Acroreader 6.0,
what is
> the problem?
No idea; I tried it, it works \fine for me <g> with Acro 4.05.
No problem, glad to have been of help,
Jentje Goslinga
===
Subject: Re: Jac(O), O = integers in a number \field
: Assume K/Q is a number \field, and let O denote the integers
in this
: extension.
: Of course, O is a Noetherian, integrally closed, domain of
(Krull)
: dimension 1, aka Dedekind domain. In particular, every
nonzero prime
: ideal is maximal and so the only nonmaximal prime ideal in
O is the
: zero ideal.
: My question: what is the Jacobson radical, Jac(O)?
Use the fact that the intersection of all prime ideals of a
commutative
ring is equal to the ideal of nilpotent elements in the ring.
Ted Hwa
===
Subject: Re: Jac(O), O = integers in a number \field
> : Assume K/Q is a number \field, and let O denote the
integers in this
> : extension.
> : Of course, O is a Noetherian, integrally closed, domain
of (Krull)
> : dimension 1, aka Dedekind domain. In particular, every
nonzero prime
> : ideal is maximal and so the only nonmaximal prime ideal
in O is the
> : zero ideal.
> : My question: what is the Jacobson radical, Jac(O)?
> Use the fact that the intersection of all prime ideals of a
commutative
> ring is equal to the ideal of nilpotent elements in the
ring.
> Ted Hwa
I do not see how this helps--although what you say is
certainly
correct, it applies to all commutative rings, even those
where the
Nilradical is *strictly* contained in the Jacobson radical.
Here,
however, I wish to prove that the Nilradical is equal to the
Jacobson
radical. Can you give some more details please?
Jenny
===
Subject: Re: Jac(O), O = integers in a number \field
>>
>> : Assume K/Q is a number \field, and let O denote the
integers in this
>> : extension.
>>
>> : Of course, O is a Noetherian, integrally closed, domain
of (Krull)
>> : dimension 1, aka Dedekind domain. In particular, every
nonzero prime
>> : ideal is maximal and so the only nonmaximal prime ideal
in O is the
>> : zero ideal.
>>
>> : My question: what is the Jacobson radical, Jac(O)?
>> Use the fact that the intersection of all prime ideals of
a commutative
>> ring is equal to the ideal of nilpotent elements in the
ring.
>> Ted Hwa
>I do not see how this helps--although what you say is
certainly
>correct, it applies to all commutative rings, even those
where the
>Nilradical is *strictly* contained in the Jacobson radical.
Here,
>however, I wish to prove that the Nilradical is equal to the
Jacobson
>radical. Can you give some more details please?
I think you can solve your original problem as follows.
Let a be a nonzero element of O. Choose a prime number p (in
Z) not
dividing
N(a), and let P be a prime ideal of O containing <p>. Then
the norm of P
(i.e. the index of P in O) is a power of p, whereas the index
of <a> in O
is coprime to p, so <a> cannot be contained in P, and hence a
is not in P.
Derek Holt.
===
Subject: Re: Ring of continuous real functions
> The expression Ôalgebrais non-descriptive, \
even
misleading,
> while a proper description Ôvector ringis \
clear and
helpful.
Not true. The expression Ôalgebrais \
overloaded, I give you
that:)
However, it is hardly ever misleading in a given context.
OTOH, the
term Ôvector ringwould lead to confusion: what \
do you mean?
an
algebra?
Mathematics is full of overloaded terminology. While this is
not
an ideal state of affairs, the alternative would be to use
words
bearing no relation whatsoever to the vocabulary of the
natural
languages. Count the various (mathematical) meanings of the
words
Ôdegreeand \
Ônormalthat you are familiar with. Any
confusions?
Hope not!
Enjoy your algebra, and learn to live with overloaded
terminology!
Jyrki Lahtonen, Turku, Finland
===
Subject: Re: Ring of continuous real functions
Count the various (mathematical) meanings of the words
> Ôdegreeand \
Ônormalthat you are familiar with. Any
confusions?
> Hope not!
Oops! May be ÔdegreeisnÕt a \
particularly useful example. In
my
native (mathematical) Finnish we use the same word for both
Ôdegreeand Ôrank\
(in pretty much all the meanings of
mathematical
English). Now thereÕs some serious overloading:)
Jyrki
===
Subject: Re: Ring of continuous real functions
<co1e0b$7n8$1@bowmore.utu.\fi> Oops! May be \
ÔdegreeisnÕt a
particularly useful example. In my
> native (mathematical) Finnish we use the same word for both
> Ôdegreeand \
Ôrank(in pretty much all the meanings of
mathematical
> English). Now thereÕs some serious overloading:)
WhatÕs the correct spelling for this sentence.
In English the word tu has three different spellings. ;-)
A train arrived at two minuts to two and departed two minute
after two.
Another train also arrived at two minuts to two and departed
two minutes
after two.
The \first train was at the station from two to two to two two.
The
second train was at the station from two to two to two two
too.
Were those two trains too overloaded? ;-)
===
Subject: Re: Ring of continuous real functions
[snipped joke]
:) Subtle (and not so subtle) puns are mostly wasted on me
(especially when IÕm down with a ßu) so I only \
\figured out
your point late last night.
Ok, help me out, please. WhatÕs the correct word?
overladen, overlaid, something else?
Jyrki
===
Subject: Re: Ring of continuous real functions
<co40tt$128t$1@bowmore.utu.\fi> Ok, help me out, please. \
WhatÕs
the correct word?
> overladen, overlaid, something else?
overburdened, overlaid.
===
Subject: Re: Ring of continuous real functions
> overburdened, overlaid.
Jyrki
===
Subject: Re: Ring of continuous real functions
> overburdened, overlaid.
IÕm not quite sure what point William is trying to make \
here.
The verb to overload is indeed the usual English term for what
you were describing, Jyrki, especially in a technical context.
--
Jim Heckman
===
Subject: Re: Ring of continuous real functions
> IÕm not quite sure what point William is trying to make
here.
> The verb to overload is indeed the usual English term for
what
> you were describing, Jyrki, especially in a technical
context.
surprisingly (?) didnÕt have the verb to overload, so I got
rather worried. My copy of WebsterÕs Thesaurus does have it,
and
it describes the words meaning the same way as you do. My
other
worry was due to the fact that I had misremembered a (related)
term overlay used in a computer programming context.
ItÕs been 14 years since I lived in an English speaking
country,
but I still feel like I sort of know English. Therefore an
occasional
reminder that IÕm not at the level of an educated native
speaker
is needed and, indeed, good for me:)
Jyrki
===
Subject: Re: Ring of continuous real functions
===
> Subject: Re: Ring of continuous real functions
[...]
>>Remember? Oh yea, linear algebra gives \
Ôalgebraa parochial
>> meaning. Properly then isnÕt an \
Ôalgebrajust an inner
product
>> space?
> ??? No. Completely different things. Most inner product
spaces
> donÕt have amultiplication. The Ôinner \
productof two
vectors
> is a scalar for an inner product space. For algebras, the
product
> of two vectors is another vector.
> An algebra A is a vector space A with a vector product?
Not just any vector product, but a *bilinear* vector product.
I.e., x(ay + bz) = a(xy) + b(xz) and (ax + by)z = a(xz) +
b(yz)
for all x,y,z in A and a,b in F. I already went over this with
you in alt.math.undergrad some time ago.
> R^3 with the cross product is an algebra?
> An algebra with a non-associative, anti-commutative product?
Of course. R^3 with the cross product is just the Lie algebra
so(3). A Lie algebra is one satisfying x^2 = 0 and the Jacobi
identity x(yz) + y(zx) + z(xy) = 0.
> Or is an algebra A is a vector space A with a vector
product that is
> both sides distributive over addition? In other words a
vector ring.
Ring usually implies associativity, which is not part of most
peopleÕs default de\finition of an algebra, \
although when
dealing
with associative algebras many authors will say something like
associative algebra.
> Four dimensional vectors with matrix multiplication is an
example
> of a non-commutative vector ring.
> For vectors x,y and scalar a, a needed axiom is
> a(xy) = (ax)y = x(ay) ?
Just a special case of bilinearity. (See de\finition above.)
> Is a (vector) multiplicative identity required?
No. But again many authors, when dealing with such algebras,
multiplicative identity [or 1].
> The expression Ôalgebrais non-descriptive, \
even
misleading,
Not if itÕs well de\fined, which it is.
> while a proper description Ôvector ringis \
clear and
helpful.
See above.
--
Jim Heckman
===
Subject: Re: Ring of continuous real functions
===
Subject: Re: Ring of continuous real functions
William Elliot <marsh@privacy.net> An algebra A is a vector
space A with a vector product?
> Not just any vector product, but a *bilinear* vector
product.
> I.e., x(ay + bz) = a(xy) + b(xz) and (ax + by)z = a(xz) +
b(yz)
> for all x,y,z in A and a,b in F. I already went over this
with
> you in alt.math.undergrad some time ago.
You did? No did \find.
> R^3 with the cross product is an algebra?
> Of course. R^3 with the cross product is just the Lie
algebra
> so(3). A Lie algebra is one satisfying x^2 = 0 and the
Jacobi
> identity x(yz) + y(zx) + z(xy) = 0.
A noncommutative, nonassociative algebra.
> Or is an algebra A is a vector space A with a vector
product that
> is both sides distributive over addition? In other words a
vector
> ring.
> Ring usually implies associativity, which is not part of
most
> peopleÕs default de\finition of an algebra, \
although when
dealing
> with associative algebras many authors will say something
like
> associative algebra.
So distributivity is the only assured ring property.
> For vectors x,y and scalar a, a needed axiom is
> a(xy) = (ax)y = x(ay) ?
> Just a special case of bilinearity. (See de\finition above.)
Ok, howÕs this? Have I got it straight?
An algebra is a vector space with a bilinear vector product.
That is, for all vectors x,y,z, scalars a,
a(xy) = (ax)y = x(ay)
x(y + z) = xy + xz, (x + y)z = xz + yz
Matrices for example are an
associative, noncommutative algebra with identity.
If F is a \field, then F, F[x], F[[x]], F(x) are F-algebras
with scalars F.
They are also algebras with scalars the prime sub\field or any
sub\field of
F.
----
===
Subject: Re: Ring of continuous real functions
[...]
> Ok, howÕs this? Have I got it straight?
> An algebra is a vector space with a bilinear vector product.
> That is, for all vectors x,y,z, scalars a,
> a(xy) = (ax)y = x(ay)
> x(y + z) = xy + xz, (x + y)z = xz + yz
Yes.
> Matrices for example are an
> associative, noncommutative algebra with identity.
*Square* matrices over a *\field*, yes. The algebra of nxn
matrices over the \field F, often denoted by M_n(F) or M(n,F),
is
extremely important in the theory of associative algebras,
since
it can shown that every n-dimensional associative algebra A
over
F is isomorphic to a subalgebra of M(n,F) if A has an
identity,
or of M(n+1,F) if not.
> If F is a \field, then F, F[x], F[[x]], F(x) are F-algebras
with scalars
F.
> They are also algebras with scalars the prime sub\field or
any sub\field of
> F.
Yes.
--
Jim Heckman
===
Subject: Ring of continuous real functions
> An algebra is a vector space with a bilinear vector product.
> That is, for all vectors x,y,z, scalars a,
> a(xy) = (ax)y = x(ay)
> x(y + z) = xy + xz, (x + y)z = xz + yz
> Matrices for example are an
> associative, noncommutative algebra with identity.
> *Square* matrices over a *\field*, yes. The algebra of nxn
> matrices over the \field F, often denoted by M_n(F) or
M(n,F), is
> extremely important in the theory of associative algebras,
since
> it can shown that every n-dimensional associative algebra A
over
> F is isomorphic to a subalgebra of M(n,F) if A has an
identity,
> or of M(n+1,F) if not.
So either way itÕs isomorphic to a subalgebra of M(n+1,F)
A subalgebra is a + and * closed subset of the algebra with
the same scalars and of course closed by scalar
multiplication.
Examples of in\finite dimensional associative algebras
are C(R) and { f:R -> R } with pointwise *.
===
Subject: Re: This WeekÕs Finds in Mathematical Physics (Week
209)
> This WeekÕs Finds in Mathematical Physics - Week 209
> John Baez
> Time ßies! This June, Peter May and I organized a workshop
on
> n-categories at the Institute for Mathematics and its
Applications:
Do you consider category theory to be a branch of physics???
Or where is the physics in This WeekÕs Finds?
Arnold Neumaier
===
Subject: Re: This WeekÕs Finds in Mathematical Physics (Week
209)
>> This WeekÕs Finds in Mathematical Physics - Week 209
>> John Baez
>> Time ßies! This June, Peter May and I organized a workshop
on
>> n-categories at the Institute for Mathematics and its
Applications:
> Do you consider category theory to be a branch of physics???
> Or where is the physics in This WeekÕs Finds?
Who cares? Category Theory is good mathematics, so
please keep it coming on sci.math.
--
Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html
Lacan, Jacques, 79, 91-92; mistakes his penis for a square
root, 88-9
Francis Wheen, _How Mumbo-Jumbo Conquered the World_
===
Subject: Re: Parametrizations of U(2)
>U(2) (the group of unitary 2x2 matrices) is a smooth
manifold of
>dimension 4 (over the real numbers), ie. there exist an
atlas of (local)
>diffeomorphisms mapping some subset of R^4 into U(2), such
that U(2) is
>completely covered.
>For example the subgroup SU(2) can be parametrized with
>{{z_1, z_2},{-(z_2)*,z_1*}} s.t. |z_1|^2 + |z_2|^2 = 1,
>where z_1 and z_2 are complex numbers. IÕm looking for
parametrizations
>of the remaining unitary matrices in U(2)SU(2) with four real
parameters.
> WhatÕs wrong with the de\finition? A matrix M = \
{{z_1,
z_2},{z_3,z_4}}
> lies in U(2) iff M* M = I, i.e.
> (z_1)* (z_1) + (z_3)* (z_3) = 1
> (z_1)* (z_2) + (z_3)* (z_4) = 0
> (z_2)* (z_1) + (z_4)* (z_3) = 0
> (z_2)* (z_2) + (z_4)* (z_4) = 1
> The middle two equations say the same thing: roughly, that
the ratios
> (z_4)/(z_1)* and -(z_2)/(z_3)* are equal, to r , say. That
is,
> z_4 = r (z_1)*, z_2 = -r (z_3)* . The \first and last
equations together
> then say that |r| = 1. So now you have a parameterization
> {{z_1, -r (z_3)*},{z_3, r(z_1)*}} s.t. |z_1|^2 + |z_2|^2 =
1, |r| = 1.
> Your parameterization of SU(2) is contained in this; SU(2)
is the
> set of matrices with r = 1.
Dave,
Andor
===
Subject: Re: the modesty of Mr Wolfram. On the Foundation of
mathematics
and
mathematica
> [ ... ] I mean, _I_ could easily have
> written Mathematica if IÕd felt like it [ ... ]
No, you couldnÕt.
Han de Bruijn
===
Subject: A restatement of the anti-Wolfram argument
> [ ... ] I mean, _I_ could easily have
> written Mathematica if IÕd felt like it [ ... ]
> No, you couldnÕt.
Han has a good point. If you tried to write something
remotely similar
to MATLAB or Mathematica from scratch you would wind up
spending long
years implementing various numerical and symbolic algorithms,
designing a language, UI, etc. It would take years, really,
even for
the best programmer on earth.
That is a counter-argument to WolframÕs work. Chaitin \
praises
Mathematica, he says itÕs a substantial AI for mathematics. \
I
agree
with Chaitin in that Mathematica knows a good deal about
mathematics,
but it probably lacks anything that could be called
mathematical
*reasoning*.
The problem is WolframÕs own argument that reality is in \
fact
based on
very simple algorithms. Why is Mathematica so complex then?
Maybe,
itÕs because mathematics is no simple thing? And if so, \
since
mathematics is part of human reality, then it turns out that
Wolfram
was wrong in his argument for universal simplicity, that
complexity is
an illusion. Complexity is a real thing that our minds are
made to
deal with. Our brains arenÕt there for nothing. And this is
easy to
see: the shortest program that can implement something like
Mathematica on a barebones universal computer is still quite
long!
--
Eray Ozkural
===
Subject: Re: A restatement of the anti-Wolfram argument
,
...... then it turns out that Wolfram was wrong in his
argument for
universal simplicity, that complexity is an illusion.
In no part of A New Kind of Science, Wolfram suggests that
complexity is an illussion.
His thesis is that complexity arises from the interaction of
simple
laws. In the same sense that the mechanics of the Solar
System is
governed by the 3 laws of Galileo and the \
ÔNewtonÕs law of
Gravitation.
His book is a disprover of the false ChaitinÕs mesure of
complexity:
The complexity of a sequence of numbres is mesured by the
length of
the minimum program that can reporduce the sequence
Here is a program that contradicts that thesis:
T = -3.14 : D = .01 : Y = 0
DO WHILE T < 3.14
X = T + SIN(5*Y)
Y = T - COS(2*X)
DRAW(X,Y)
T = T + D
LOOP
This is a curve perfectly simetric and periodic, that is of
low
complexity.
But if you change the Y for X in the third line and de X foy
in the
fourth, it results a chaotic curve without periodicity.
The length of the two programs are the same but not its
complexity.
Refer.
G. CHAITIN Randomnes ans Mathematical Proof Scienti\fic \
American
1975
Exploring Randomness Springer 2001
I. STEWART Does God Play Dice? Blackwell 1989
===
Subject: Re: A restatement of the anti-Wolfram argument
> The problem is WolframÕs own argument that reality is in
fact based on
> very simple algorithms. Why is Mathematica so complex then?
Maybe,
> itÕs because mathematics is no simple thing?
WolframÕs point seems to be that all that complex \
mathematics
is
redundant: in order to effectively understand and manipulate
the real
world, much simpler ideas might suf\fice, or even be more
useful. (again:
WÕs point, not mine).
And if so, since
> mathematics is part of human reality,
Perhaps -human- reality is not what W. is talking about.
then it turns out that Wolfram
> was wrong in his argument for universal simplicity, that
complexity is
> an illusion. Complexity is a real thing that our minds are
made to
> deal with. Our brains arenÕt there for nothing. And this \
is
easy to
> see: the shortest program that can implement something like
> Mathematica on a barebones universal computer is still
quite long!
Perhaps Mathematica is complex only because -humans- have
made their own
lives and thoughts too complex (in other words, because
mathematics is
unnecessarily complex). (Again: attempts to present WÕs pov,
not my pov).
--
Herman Jurjus
===
Subject: Re: A restatement of the anti-Wolfram argument
> The problem is WolframÕs own argument that reality is in
fact based on
> very simple algorithms. Why is Mathematica so complex then?
Maybe,
> itÕs because mathematics is no simple thing?
> WolframÕs point seems to be that all that complex
mathematics is
> redundant: in order to effectively understand and
manipulate the real
> world, much simpler ideas might suf\fice, or even be more
useful. (again:
> WÕs point, not mine).
> And if so, since
> mathematics is part of human reality,
> Perhaps -human- reality is not what W. is talking about.
> then it turns out that Wolfram
> was wrong in his argument for universal simplicity, that
complexity is
> an illusion. Complexity is a real thing that our minds are
made to
> deal with. Our brains arenÕt there for nothing. And this \
is
easy to
> see: the shortest program that can implement something like
> Mathematica on a barebones universal computer is still
quite long!
> Perhaps Mathematica is complex only because -humans- have
made their own
> lives and thoughts too complex (in other words, because
mathematics is
> unnecessarily complex). (Again: attempts to present WÕs
pov, not my pov).
Yes, I understand W.s point, but I also think heÕs
overlooking the
enormous amount of complexity created by billions of years of
evolution. I donÕt think Wolfram understands that the life
processes
are very detailed and complex processed that cannot really be
dumbed
down to 1D CAs. Every few years, we \find new stuff in
neuroscience and
molecular biology that suggests that we should multiply the
complexity
of some things we thought were simple by ten.
Regardless of experimental \findings, if W. were right, then it
should
have been the case that Mathematica is really only seemingly
complex,
and its real complexity, program-size complexity should have
been
extremely small. But by any stretch of imagination, you \
canÕt
compress, with current human knowledge, all of Mathematica to
less
than a hundred kilobytes, and that is a remarkable
complexity. (Maybe
the absolute compressibility is much larger, probably so, but
itÕs not
a Rule 110 or anything like that)
Also for the brain, W. seems to suggest that the brain is the
result
of a very very tiny program, which I \find myself in sheer
disagreement
with. I am certain that it is optimized very well, but itÕs
not 10
bits.
And one other point, surely a {0,1}* generator can create all
the
complexity in the world. But that takes time and space! An
inordinate
amount of time and space! It is COMPLETELY IRRELEVANT to
claim that
since there is such a generator, COMPLEXITY is an ILLUSION. I
would
not write a volume to explain the previous two sentences!
What a waste
of resources!
--
Eray
===
Subject: Re: A restatement of the anti-Wolfram argument
<cnuft4$pho$1@news.math.niu.edu>
<see6q0phb8n8c5169v3m992q7i5n5b1l9r@4ax.com>
<co1era$mhp$1@news.tudelft.nl>
Discussion, linux)
>> [ ... ] I mean, _I_ could easily have
>> written Mathematica if IÕd felt like it [ ... ]
>> No, you couldnÕt.
> Han has a good point.
He might have a good point, if anyone with two brain cells to
rub
together thought David was serious.
--
Jesse F. Hughes
[Iota]Õs the smallest in\finitesimal, Russell, \
there are smaller
in\finitesimals. -- Ross Finlayson
===
Subject: Re: A restatement of the anti-Wolfram argument
>>
>> [ ... ] I mean, _I_ could easily have
>> written Mathematica if IÕd felt like it [ ... ]
>>
>> No, you couldnÕt.
> Han has a good point.
> He might have a good point, if anyone with two brain cells
to rub
> together thought David was serious.
Well, then letÕs see what David says. That statement was
taken out of
quote, he was really poking fun about some other issue, but
there is
no reason why he should not be half-serious.
--
Eray Ozkural
===
Subject: Re: A restatement of the anti-Wolfram argument
<cnuft4$pho$1@news.math.niu.edu>
<see6q0phb8n8c5169v3m992q7i5n5b1l9r@4ax.com>
<co1era$mhp$1@news.tudelft.nl>
<878y8pvhx2.fsf@phiwumbda.org>
Discussion, linux)
>
> [ ... ] I mean, _I_ could easily have
> written Mathematica if IÕd felt like it [ ... ]
>
> No, you couldnÕt.
> Han has a good point.
>> He might have a good point, if anyone with two brain cells
to rub
>> together thought David was serious.
> Well, then letÕs see what David says. That statement was
taken out of
> quote, he was really poking fun about some other issue, but
there is
> no reason why he should not be half-serious.
Maybe itÕs simply a bit of cultural knowledge, but I believe
any
construction of the form I could easily have do [some
monumental
task] if IÕd felt like it is a joke.
Unless James Harris says it.
Of course, we all have our suspicions about James S Harris ==
David
C. Ullrich, so he might surprise me here.
--
I arrest anybody I think needs arresting, Mr. Carter, and \
IÕm
not in
the habit of explaining why.
ThereÕs a law about that ---
YouÕre in Dodge, Mr. Carter. -- Gunsmoke radio show (A Bush
favorite)
===
Subject: Re: the modesty of Mr Wolfram. On the Foundation of
mathematics and
mathematica
~^>Pn0&%&Ux8>1=w8P?^q%:g?%]2+oVLC;x!s,~MYjl!j>x`k>b9B5_NaMÕ4_\

X:z
Zw76--
> I asked here once, does Wolfram have any important theorem
whatsoever
> about cellular automata, or all he has is a handful of CA
runs? The
> important theorems I remember were found by some other
people, maybe
> itÕs my memory that does not serve me right.
> IÕm asking this, because I want to decide if I should buy
the book,
> and that is the litmus test for me. If he has any \
signi\ficant
> theorems, then the book would be worth reading. Otherwise,
itÕs good
> for knocking out your roommate. I saw it. ItÕs thick, and
it has
> pretty pictures, but thatÕs not suf\ficient for \
me.
If your criteria for buying NKS is it needs to have some new
important
theorem in it, then you shouldnÕt buy it. But this criteria
would
eliminate a fairly sizable portion of quite useful books.
I can think of a couple of good reasons to buy the book.
First, if you
have an interest in cellular automata, NKS covers a wide
variety of
cellular automata and shows examples of how they can be
related to a
fairly wide variety of problems. So, NKS is a quite useful
reference for
anyone interested in cellular automata.
Second, whether you accept the concepts presented in NKS as a
new kind
of science or not they clearly represent a different way of
thinking
about a variety. And it is quite useful to having a fresh
viewpoint
whether you agree with the viewpoint or not. Simply thinking
about why
an argument is or is not valid is quite useful.
But if you canÕt get past WolframÕs style in \
this book, canÕt
stand
Mathematica or insist on rigorous proof of all concepts
presented then
you will de\finitely not be happy with this book.
--
To reply via email subtract one hundred nine
===
Subject: Re: the modesty of Mr Wolfram. On the Foundation of
mathematics and
mathematica
> IÕm asking this, because I want to decide if I should buy
the book,
> and that is the litmus test for me. If he has any \
signi\ficant
> theorems, then the book would be worth reading. Otherwise,
itÕs good
> for knocking out your roommate. I saw it. ItÕs thick, and
it has
> pretty pictures, but thatÕs not suf\ficient for \
me.
> If your criteria for buying NKS is it needs to have some
new important
> theorem in it, then you shouldnÕt buy it. But this \
criteria
would
> eliminate a fairly sizable portion of quite useful books.
> I can think of a couple of good reasons to buy the book.
First, if you
> have an interest in cellular automata, NKS covers a wide
variety of
> cellular automata and shows examples of how they can be
related to a
> fairly wide variety of problems. So, NKS is a quite useful
reference for
> anyone interested in cellular automata.
Yes, IÕm interested in CAs.
> Second, whether you accept the concepts presented in NKS as
a new kind
> of science or not they clearly represent a different way of
thinking
> about a variety. And it is quite useful to having a fresh
viewpoint
> whether you agree with the viewpoint or not. Simply
thinking about why
> an argument is or is not valid is quite useful.
In fact, I agree that a computational POV is useful.
> But if you canÕt get past WolframÕs style in \
this book,
canÕt stand
> Mathematica or insist on rigorous proof of all concepts
presented then
> you will de\finitely not be happy with this book.
I started reading the online version but quickly grew
suspicious of
the content, and I couldnÕt continue reading because the
online
version hurt my eyes. I think, from such a title, I would be
expecting
a really major result or demonstration of some computational
phenomena
that would make me rethink everything. I might even agree
that a
purely experimental approach can be useful. Maybe he should
just
compress the relevant ideas minus the ranting to a 40-page
paper.Compression is a sign of intelligence.
--
Eray Ozkural
===
Subject: Re: the modesty of Mr Wolfram. On the Foundation of
mathematics and
mathematica
>Say, now that people are talking about it, has anyone ever
seen Mr.W
>and JSH in the same room at the same time? Hm...
> Hmm, indeed. Some might suggest that JSHÕs bitterness \
about
the fact
> that mathworld refuses to mention his work shows that your
unstated
> conjecture is full of beans. But itÕs clear to me that
thatÕs all
> just a smokescreen.
> Enhances oneÕs respect for Wolfram. I mean, _I_ could
easily have
> written Mathematica if IÕd felt like it, but I \
canÕt
imitate JSHÕs
> writing style - IÕve tried many times.
Yeah right.
YouÕre not fooling anyone David.
I mean is anyone stupid enough to believe that JSH
is for real?
Now we need someone with the mathematical and
technical knowledge to become JSH. Also this
person has to have been hanging around net^H^H^Hsci.math
for long enough. Now consider:
David Ullrich Started posting Feb 1995
James Harris Started posting Feb 1996
Both are still around.
What more evidence do we need?
-William Hughes
===
Subject: Re: Is absolute integral really the neccessary and
suf\ficient
condition for a system to be stable?
>> For Bounded-In-Bounded-Output system,
>> Is the absolute integrability the neccessary and suf\ficient
condition for
>> a
>> system to be stable?
>> ThatÕs to say: Integrate(abs(h(t)), t from -inf to inf) <
+inf
>> Any proof?
>> I also want to know if this is the condition for LTI
system only, or it
>> is
>> the condition for Linear systems or Time Invariant systems.
>> Absolute integral is dif\ficult to check, does it have any
alternative
>> equivalent condtions?
> The very word impulse response, h(t) exists only for LTI
systems. so
> your formula for stability applies to LTI. The equivalent
is :\finding
> zero input response of system i.e characteristic roots and
modes. I
> think thats universal i.e for stability of any system,
charateristic
> mode should --->0 when t--->inf.
> A. Kumar
Hi Kumar,
modes for a system? and for (non)linear system?
===
Subject: question about stablility of systems
Hi all,
About the criteria of systems stablity:
1) h(t) absolutely integrable, for LTI systems.
2) the eigenvalues having negative real part(continous
system). (for LTI
systems only???)
3) the eigenvalues having magnitude less than 1(discrete
system). (for LTI
systems only???)
what else criteria do we have?
I am also wondering which of these critiria are applicable to
LTI system
only, which are applicable to Linear system, and which are
applicable to
non-linear systems...
===
Subject: Re: question about stablility of systems
I did not mean that Lyapunov is not used for Linear Systems.
I mean to say is, there is no need to take Lyapunov Function
for Small
Linear
Systems. I appreciate you for suggestion Lyapunov Criteria.
> Hi all,
> About the criteria of systems stablity:
> 1) h(t) absolutely integrable, for LTI systems.
> 2) the eigenvalues having negative real part(continous
system). (for LTI
> systems only???)
> 3) the eigenvalues having magnitude less than 1(discrete
system). (for
LTI
> systems only???)
> what else criteria do we have?
> I am also wondering which of these critiria are applicable
to LTI system
> only, which are applicable to Linear system, and which are
applicable to
> non-linear systems...
===
Subject: Re: question about stablility of systems
4) It should be BIBO stable( Bounded Input and Bounded
Output) says, for \finte input the systme output should be
\finite.
Athreya
> Hi all,
> About the criteria of systems stablity:
> 1) h(t) absolutely integrable, for LTI systems.
> 2) the eigenvalues having negative real part(continous
system). (for LTI
> systems only???)
> 3) the eigenvalues having magnitude less than 1(discrete
system). (for
LTI
> systems only???)
> what else criteria do we have?
> I am also wondering which of these critiria are applicable
to LTI system
> only, which are applicable to Linear system, and which are
applicable to
> non-linear systems...
===
Subject: Re: question about stablility of systems
Negative real parts correspond to exponential decay, e^(-at).
Any function with such a characteristic will result in
transient
changes dying away.
Positive real parts correspond to exponential growth, e^(+at).
Any function with such a characteristic will result in
transient
changes building up.
This is a characteristic of all systems. However, if a
non-linear
system has bounds (BIBO) then any instability resulting from
positive real will be limited also.
> 2) the eigenvalues having negative real part(continous
system). (for LTI
> systems only???)
===
Subject: why is an image non-stationary?
Hi all,
I am not sure I completely understood...
but IÕve heard people say that an image is non-stationary,
blah blah blah,
what does that mean?
and what does that imply?
I vaguely heard that people say since an image is
not-stationary, so Fourier
Transform should not be applied, etc...
Can anybody throw some lights to me?
===
Subject: Re: why is an image non-stationary?
> Hi all,
> I am not sure I completely understood...
> but IÕve heard people say that an image is non-stationary,
blah blah
blah,
> what does that mean?
> and what does that imply?
> I vaguely heard that people say since an image is
not-stationary, so
Fourier
> Transform should not be applied, etc...
Fourier analysis is based on two assumptions. One is that the
system is
superposable, usually just called linear. The other is that
the system
is time invariant in that the origin of time does not matter
for the
For images one uses position rather than time. Superposable
is OK.
Position invariance would mean that a seascape would have the
same
properties as an urban image. If you think not then you have
given
up the Fourier assumptions. The things folks will agree on
tend to
suggest Haar analysis or perhaps wavelets.
> Can anybody throw some lights to me?
===
Subject: Re: why is an image non-stationary?
> Position invariance would mean that a seascape would have
the same
> properties as an urban image. If you think not then you
have given
> up the Fourier assumptions. The things folks will agree on
tend to
> suggest Haar analysis or perhaps wavelets.
I actually think itÕs a little different from the way you \
say
it. I
donÕt think itÕs that the stationarity problem \
occurs between
different images, itÕs that the statistics of an image \
change
_within_
a given image.
For example, the pixel values of the sea in a seascape would
have
different mean and standard deviation from the pixel values
of the
beach in the same seascape.
ThatÕs why images can be thought of as non-stationary.
For what itÕs worth, I really donÕt think that \
stationarity
says
anything about whether the Fourier transform can or canÕt be
used. Of
course it can be used; how you interpret it might be a
problem, but
many non-stationary problems (e.g. speech, sonar) have used
the
Fourier transform to good effect.
Ciao,
Peter K.
===
Subject: Re: why is an image non-stationary?
>> I am not sure I completely understood...
>> but IÕve heard people say that an image is \
non-stationary,
blah blah
blah,
>> what does that mean?
[I was sort of hoping that a mathematician would chime in
here, because
I had noticed a spate of sci.math threads about these
signal-processing
applications, all very confusing to me: they appear to use
mathematiciansÕ
terminology (linear, etc.) to mean something else.]
>Fourier analysis is based on two assumptions. One is that
the system is
>superposable, usually just called linear. The other is that
the system
>is time invariant in that the origin of time does not matter
for the
>For images one uses position rather than time. Superposable
is OK.
>Position invariance would mean that a seascape would have
the same
>properties as an urban image. If you think not then you have
given
>up the Fourier assumptions. The things folks will agree on
tend to
>suggest Haar analysis or perhaps wavelets.
Um, right. THere has to be a more rigorous way to put this!
Sure,
sound \filters work the same at morning and night. And image
\filters
work the same in London as in Hong Kong. So whatÕs the \
point?
I think what you _meant_ to say was that the Fourier series
can be computed
over any length of time (if itÕs a multiple of the period of
the signal);
you donÕt need to know what the starting point of a period \
is.
But the same could be said of an image IF itÕs periodic. You
could
for example compute a fourier series for the background
images on many
computer screens --- the ones that tile endlessly. Just as
with the
sound \filter, you could start your computations anywhere
within the
tile (fundamental domain, in math parlance) and get the same
answers.
What prompts the use of wavelets and other things is
precisely the
lack of periodicity. That would be true of sound \filters too:
it would
be pointless to try to view the sound wave of, say, a
one-hour conversation
as if it were a complex superposition of sound waves with
periods which
evenly divided 1 hour!
dave
(writing from sci.math)
===
Subject: Re: why is an image non-stationary?
>I am not sure I completely understood...
>but IÕve heard people say that an image is non-stationary,
blah blah
blah,
>what does that mean?
> [I was sort of hoping that a mathematician would chime in
here, because
> I had noticed a spate of sci.math threads about these
signal-processing
> applications, all very confusing to me: they appear to use
mathematiciansÕ
> terminology (linear, etc.) to mean something else.]
Partially there needs to be a translation guide, partially
some of the
terms are used loosely. Linear System, speci\fically, causes
confusion.
I think that when a mathematician sees the phrase he thinks
linear
system of equations. The signal & systems person, however,
sees this
and thinks linear dynamical system, in which a system is a
thingie
that transforms one signal into another, a dynamical system
is one where
the current value of the output signal is a function of the
history of
the input signal (possibly only the past history, possibly
past, present
& future), and a linear dynamical system is one that obeys
superposition, i.e. the output signal that results from the
sum of two
input signals is exactly equal to the sum of the output
signals that
would have resulted from the two input signals processed
individually.
>>Fourier analysis is based on two assumptions. One is that
the system is
>>superposable, usually just called linear. The other is that
the system
>>is time invariant in that the origin of time does not
matter for the
>>For images one uses position rather than time. Superposable
is OK.
>>Position invariance would mean that a seascape would have
the same
>>properties as an urban image. If you think not then you
have given
>>up the Fourier assumptions. The things folks will agree on
tend to
>>suggest Haar analysis or perhaps wavelets.
> Um, right. THere has to be a more rigorous way to put this!
Sure,
> sound \filters work the same at morning and night. And image
\filters
> work the same in London as in Hong Kong. So whatÕs the
point?
> I think what you _meant_ to say was that the Fourier series
can be
computed
> over any length of time (if itÕs a multiple of the period
of the signal);
> you donÕt need to know what the starting point of a period
is.
> But the same could be said of an image IF itÕs periodic.
You could
> for example compute a fourier series for the background
images on many
> computer screens --- the ones that tile endlessly. Just as
with the
> sound \filter, you could start your computations anywhere
within the
> tile (fundamental domain, in math parlance) and get the
same answers.
> What prompts the use of wavelets and other things is
precisely the
> lack of periodicity. That would be true of sound \filters
too: it would
> be pointless to try to view the sound wave of, say, a
one-hour
conversation
> as if it were a complex superposition of sound waves with
periods which
> evenly divided 1 hour!
> dave
> (writing from sci.math)
I have a question myself about the use of the word stationary
that IÕm
going to post separately.
--
Tim Wescott
Wescott Design Services
http://www.wescottdesign.com
===
Subject: Re: why is an image non-stationary?
>I am not sure I completely understood...
>but IÕve heard people say that an image is non-stationary,
blah blah
blah,
>what does that mean?
> [I was sort of hoping that a mathematician would chime in
here, because
> I had noticed a spate of sci.math threads about these
signal-processing
> applications, all very confusing to me: they appear to use
mathematiciansÕ
> terminology (linear, etc.) to mean something else.]
Good mathematicians are able to use the terminology of their
applications if that helps communications. Linear is so
pervasive
that it has many detailed technical meanings when it is not
embedded in much longer unambiguous technical phrases.
>>Fourier analysis is based on two assumptions. One is that
the system is
>>superposable, usually just called linear. The other is that
the system
>>is time invariant in that the origin of time does not
matter for the
>>For images one uses position rather than time. Superposable
is OK.
>>Position invariance would mean that a seascape would have
the same
>>properties as an urban image. If you think not then you
have given
>>up the Fourier assumptions. The things folks will agree on
tend to
>>suggest Haar analysis or perhaps wavelets.
> Um, right. THere has to be a more rigorous way to put this!
Sure,
> sound \filters work the same at morning and night. And image
\filters
> work the same in London as in Hong Kong. So whatÕs the
point?
Sound \filters work the same if the time origin is relabelled,
otherwise
known as time invariance. Images do not have position
invariance even
if the same processing might be applied in both NYC and HK.
Depending
upon the invariance conditions you get differing sorts of a
diagonalizing
transformation for the operators.
> I think what you _meant_ to say was that the Fourier series
can be
computed
> over any length of time (if itÕs a multiple of the period
of the signal);
Gee! I was not aware that the usual integral Fourier
transform de\fined
on the real line had any requirements that the function under
analysis be
periodic. If the indexing group is periodic then you get what
is usually
called Fourier Series with its discrete frequencies. What was
meant is
that the Fourier transform is a diagonalizing operation for
operators
which are superposable, linear in the common jargon, and time
invariant
with respect to their time index whether it be unbounded or
periodic,
continuous or discrete. There being slightly differing FTs
for the
various underlying indexing groups.
> you donÕt need to know what the starting point of a period
is.
> But the same could be said of an image IF itÕs periodic.
You could
But images are not periodic. Compression schemes based on
periodic
assumptions tend to particular styles of artifacts. Often
they are made
periodic by \first reßecting and then repeating to generate
only even
Fourier coef\ficients, i.e. cosines.
> for example compute a fourier series for the background
images on many
> computer screens --- the ones that tile endlessly. Just as
with the
> sound \filter, you could start your computations anywhere
within the
> tile (fundamental domain, in math parlance) and get the
same answers.
> What prompts the use of wavelets and other things is
precisely the
> lack of periodicity. That would be true of sound \filters
too: it would
> be pointless to try to view the sound wave of, say, a
one-hour
conversation
> as if it were a complex superposition of sound waves with
periods which
> evenly divided 1 hour!
> dave
> (writing from sci.math)
===
Subject: Re: why is an image non-stationary?
>>Fourier analysis is based on two assumptions. One is that
the system is
>>superposable, usually just called linear. The other is that
the system
>>is time invariant in that the origin of time does not
matter for the
>>For images one uses position rather than time. Superposable
is OK.
>>Position invariance would mean that a seascape would have
the same
>>properties as an urban image. If you think not then you
have given
>>up the Fourier assumptions. The things folks will agree on
tend to
>>suggest Haar analysis or perhaps wavelets.
> Um, right. THere has to be a more rigorous way to put this!
Sure,
> sound \filters work the same at morning and night. And image
\filters
> work the same in London as in Hong Kong. So whatÕs the
point?
Time and position invariance. There is a close relation
between
symmetry and conservation laws. Conservation of energy is
related
to time invariance, conservation of momentum to position
invariance.
Conservation of angular momentum to rotation invariance.
Note also that energy*time, momentum*distance, and angular
position
all have the same dimensions.
(snip)
-- glen
===
Subject: Re: why is an image non-stationary?
> Fourier analysis is based on two assumptions. One is that
the system is
> superposable, usually just called linear. The other is that
the system
> is time invariant in that the origin of time does not
matter for the
> For images one uses position rather than time. Superposable
is OK.
> Position invariance would mean that a seascape would have
the same
> properties as an urban image. If you think not then you
have given
> up the Fourier assumptions. The things folks will agree on
tend to
> suggest Haar analysis or perhaps wavelets.
>> Um, right. THere has to be a more rigorous way to put
this! Sure,
>> sound \filters work the same at morning and night. And image
\filters
>> work the same in London as in Hong Kong. So whatÕs the
point?
> Time and position invariance. There is a close relation
between
> symmetry and conservation laws. Conservation of energy is
related
> to time invariance, conservation of momentum to position
invariance.
> Conservation of angular momentum to rotation invariance.
> Note also that energy*time, momentum*distance, and angular
position
> all have the same dimensions.
> (snip)
> -- glen
Hi all,
Just wanted to chip in with an electrical engineerÕs
viewpoint.
First of all, describing a system as LSI - linear and
shift-invariant -
only describes the system itself. It says nothing at all
about the
characteristics of the input signal. Shift invariance thus
does NOT
imply that ...a seascape would have the same properties as an
urban
image.
If a system is shift invariant, it just means that the system
response
does not vary with time or position. Mathematically, if a
given input
f(t) produces an output g(t), then if the system is LSI, a
time-delayed
input f(t + T) produces a time-delayed output g(t + T).
For an LSI imaging system, shift invariance means that the
optical
systemÕs impulse response (point spread function) is \
constant
over the
\field of view of the system.
Modeling image formation as an LSI process is the basis of
Fourier
optics - see the excellent text by J. Goodman.
H
PS Regarding the OPÕs question on image stationarity: If the
intensity
distribution of the object being imaged is time-varying, then
a time
sequence of images of that object will also display a
time-varying
intensity, and hence the image intensity is non-stationary.
===
Subject: Re: why is an image non-stationary?
Earlier posted for wrong question.
Image is always non stationary, as the expectation of pixel
matrix is non zero one.
Athreya
> Hi all,
> I am not sure I completely understood...
> but IÕve heard people say that an image is non-stationary,
blah blah
blah,
> what does that mean?
> and what does that imply?
> I vaguely heard that people say since an image is
not-stationary, so
Fourier
> Transform should not be applied, etc...
> Can anybody throw some lights to me?
===
Subject: Re: why is an image non-stationary?
4) It should be BIBO stable( Bounded Input and Bounded
Output) says, for \finte input the systme output should be
\finite.
Athreya
> Hi all,
> I am not sure I completely understood...
> but IÕve heard people say that an image is non-stationary,
blah blah
blah,
> what does that mean?
> and what does that imply?
> I vaguely heard that people say since an image is
not-stationary, so
Fourier
> Transform should not be applied, etc...
> Can anybody throw some lights to me?
===
Subject: Re: how to \find the autocorrelation and spectrum of
the receiver
signal in mobile communication?
> Hi kiki,
> As you have a time domain model of your signal just sample
it and
> using matlab PSD function (it does the WelchÕs PSD
estimation).
> You must have signal long enought.
> r[n]=r(delta_t*n)
> psd(r[n]);
> The above is mixture of mathematics and matlab syntax ...
but I think
> you got the idea.
> Best Reagads,
> penev
> -----------------
> DSP Forum: www.dsp-bg.info
>> Hi all,
>> I am wondering about the simplest model in mobile
communicaiton,
>> multipath.
>> Suppose the received signal has random uniform [0, 2*pi]
phase due to
>> multipath fading, and also has Rayleigh distribution on its
>> amplitude(assuming no direct line path), and also has
doppler frequency
>> shift in carrier frequency.
>> The signal then can be modelled as
>> r(t)=A*exp(j*2*pi*(f+delta_f)*t + THETA)
>> where A and THETA are random variables...
>> How to \find its autocorrelation function and then Power
Spectrum
Density?
Hi Dimitra,
thatÕs great! I did not know that before... I learned a lot
from this
experiment: I did the following:
Fs=1000;
t=[0:1/Fs:20];
A=random(ÔrayleighÕ, 10, 1, length(t));
THETA=rand(1, length(t))*2*pi;
delta_f=300;
f=1000000;
r=A.*exp(1j*2*pi*(f+delta_f).*t + THETA);
h = spectrum.welch;
psd(h, r, ÔFsÕ, Fs);
----------------------------------
I saw basically a narrow peak centered at delta_f=300
position, it is indeed
very high compared with all other frequency components, which
are noisily
ßat and small...
I am not sure if my model is correct though...
Can anybody tell me if my following model is correct in
simple multipath
fading mobile communication or not?
>> Suppose the received signal has random uniform [0, 2*pi]
phase due to
>> multipath fading, and also has Rayleigh distribution on its
>> amplitude(assuming no direct line path), and also has
doppler frequency
>> shift in carrier frequency.
>> The signal then can be modelled as
>> r(t)=A*exp(j*2*pi*(f+delta_f)*t + THETA)
===
Subject: Re: SkolemÕs Paradox and why is math the way it is?
J.E.:
[...]
|> In any case, those of us who are not formalists seldom
care whether
|> the Riemann hypothesis is a theorem of ZFC or PA or
whatever, or
|> whether the twin prime conjecture is a theorem of ZFC. We
do care
|> about whether theyÕre true, however. The formalist thinks
somehow
|> that these questions are not well enough de\fined, but
everybody
|> else aside from Essenin Volpin as far as I know disagrees.
|
|You totally lost me here, people care about whether a
statement is
|true when itÕs true in some models and false in others? \
Just
|consider the models where itÕs true, now itÕs \
true. Or
consider the
|ones where itÕs false, now itÕs false. Why \
would anyone care
about
|this? Am I therefore a formalist to \find this silly?
You have to be something like a formalist, yes. But you seem
not to
be very pure in your formalism. A pure formalist doesnÕt
think of it
in terms of models, because a model is, after all, a set (or
a class),
which would imply that their actual foundation is some kind
of set
theory. But a pure formalist doesnÕt start from set theory.
And then
you start talking about IF-logic, which does not have a formal
consequence relation at all.
I suspect many troubles have been created by the belief that
the
point of view you are describing here is some kind of
consensus
view in mathematics. Far from it! Quite a lot of
mathematicians
would consider the question of whether there are models
relative
to which a sentence is true of interest *only* because of the
possible light it might shed on whether they are true with the
terms meaning what they were intended to mean.
DonÕt lose yourself in broad generalities here. Think of the
particular
mathematical questions that we have, like the twin prime
conjecture.
The twin prime pairs, 3,5; 5,7; 11,13; 17,19; 29,31; ...
appear to
continue inde\finitely. Does it really seem plausible to you
that the
answer to whether they really do could possibly be yes or no,
take
your pick, depends on which axioms you use?
[...]
|> I just went over to my bookshelf and opened a group theory
|> textbook at a random page. The theorem there was that the
center
|> of the group GL(n,F) consists of the set of diagonal
matrices.
|> GL(n,F) consists of the invertible n by n matrices with
entries
|> in the \field F.
|> What are the axioms that supposedly de\fine GL(n,F)? We all
know
|> what natural numbers are, and what invertible n by n
matricies
|> are, but not because there are axioms for them. An n by n
|> matrix is a function from {1,2,...,n}x{1,2,...,n} to F;
invertibility
|> means that there exists another such one that is its
inverse, etc.
|> The starting point is arithmetic, i.e., knowing what it
means
|> to have a natural number n. What complete axiomatization of
|> arithmetic do you have in mind when doing group theory?
|
|IÕm really confused, I took this course called abstract
algebra and
|we didnÕt assume any axiomatization of arithmetic, if fact
the whole
|point was to avoid that, but instead to axiomize a group so
that
|later anything that was a group would have the group theory
theorems
|true of it.
Certainly group is \first-order axiomatizable, but GL(n,F) has
no \first-order axiomatization.
| The group GL(n,F) satis\fies the group axiom with the
|multiplication that you would expect for it (any in fact
since F
|satis\fies the \field axioms by hypothesis, you can \
prove that
GL(n,F)
|is a group, which is GOOD because that makes the nomen group
|well-de\finedish with the fact that GL(n,F) satisifes the
group axioms,
|the whole point is that the results we proved about the
center of
|abstract groups can then be applied to the subcase of groups
GL(n,F).
|Why you think this starts with arithmetic is comletely
behind me, you
|have an abstract \field F, and from it you make a group
GL(n,F), where
|does arthemetic come in?
How did you de\fine GL(n,F) for an arbitrary natural number n,
if
you didnÕt have any concept of natural number? What do you
think
arithmetic is about, if itÕs not about natural numbers?
In your abstract algebra course, did you talk about torsion in
an abelian group? In an abelian group, the set of elements g
such
that n*g=e for some n, i.e. g+g+...+g (with n gÕs) is the
identity,
are called the torsion elements, and they form a subgroup of
the
group. But you have to be able to refer to natural numbers n
(or
something that serves a similar purpose) in order to be able
to
de\fine torsion.
Group theory is full of results that are about speci\fic kinds
of
groups, where the kind of group is not given by a set of
axioms.
|> | \field theory,
|> I donÕt really have a book just on \field \
theory as far as I
know,
|> but it occurs to me that one \field being algebraic over
another
|> isnÕt \first-order de\finable. The \
property of a \field
extension,
|> that the over\field is a *\finite* dimensional \
vector space
over
|> the sub\field, comes up often. The \finiteness \
intended is
what we
|> (foolishly?) understood as just plain \finiteness, not
\finiteness
|> relative to a model of [something].
|
|The common axiomization of \field include the term set, hence
the
|importance of the question what is a set.
A group is also usually de\fined as a set with some additional
structure. This is not a respect in which the de\finition of
\field is different from the de\finition of group.
| If you threw in F is a
|\field iff (FA1, and FA2, ... FAn, and STA1, STA2, ... , and
STAn)
|(where FAk is a \field axiom and STAk is a set theory axioms)
then
|youÕd know what the models of \fields are, but \
without, you
have to beg
|the question over to set theory and ask is this a set to
know if
|something is a model of the \field axioms.
So are you beginning to see why I donÕt buy this idea that
everything else works so wonderfully deductively until mean
olÕ
set theory comes along and screws everything up? All these
lovely
algebraic topics rely a little bit upon set theory. We would
be
equally unhappy with the claim that a GL(n,F)-object in a
model
of some theory is just as good as a real GL(n,F), as we are
with
the idea that a real-line object in a model of some theory is
just
as good as the real line.
|> | geometry, etc.
|> How many colors are needed to color the points inside a
unit
|> square, so that no two points of the same color are a
distance
|> of 1/2 apart?
|> When people doing Euclidean geometry talk about the
Euclidean
|> plane, they are talking about the one thatÕs isometric to
R^2,
|> pairs of real numbers, not an arbitrary model of some
\first-order
|> axiomatization of it.
|
|Classical geometry is both complete (every statement with the
|uninterpreted primatives or geometry is either a theorem or
the
|negation of a theorem) and categorical (all models are
isomorphic)
The theory of elementary geometry, the \first order theory
having
primitives for things like collinearity, congruence, and so
on, is
indeed complete, but I intentionally chose a question about
Euclidean
geometry that doesnÕt \fit in it.
It is not categorical. No \first-order theory that has an
in\finite model
is categorical.
|as
|an axiomatic theory, so what is arbitrary about itÕs \
models?
And
|geometry doesnÕt have a primative color. \
YouÕve lost me
completely
|again. IsnÕt that really a question about functions from
manifolds
|into integers?
Manifolds? One extremely speci\fic one.
ItÕs commonplace in combinatorics to describe a function \
from
a
domain to an arbitrary (usually \finite) set with a given \
number
of elements as a coloring. Questions about colorings of the
Euclidean plane would ordinarily be categorized as questions
in
Euclidean geometry, merely not elementary Euclidean geometry.
|> Generally, your statement comes much closer to correct if
we
|> are considering relationships between second-order
statements.
|> But thereÕs no formal deductive system for second-order
statements
|> that captures all the valid deductions that can be made in
|> second-order logic. Second-order logic also involves
referring
|> to arbitrary subsets of the domain, which is the usual
bugaboo
|> of set theory.
|
|Formal deductive methods donÕt work for set theory correct,
but they
|do for geometry I donÕt know why you think they \
donÕt.
I think I misunderstood the point of what you were saying
before.
I was giving examples of theorems in these various areas of
mathematics that arenÕt of the form, any structure \
satisfying
these axioms satis\fies this theorem too. You appeared to be
saying
that set theory alone failed to \fit into this mold.
There is little difference for the purposes of this discussion
between group theory and set theory. You indicated that
something
seemed to stop working when it came to set theory, and I
think I
misunderstood.
There is no complete formal system for Euclidean geometry,
just
for the so-called elementary part of it.
| I donÕt know
|why you keep bringing up formal deduction anyway, we donÕt
use
|deduction to design theories, so what is this prima facie
reason for
|such a hubbub about it?
Hubbub? Oh, creating a hubbub, all by myself? I didnÕt \
realize
I had it in me.
Formal deduction is relevant because of its relationship with
models.
(Essentially) the only thing that other models of ZFC have in
common
with the cumulative hierarchy is that they satisfy the
*formally
deducible consequences* of the axioms of ZFC, as relativized
to them.
If you do not require the theory to be formal, if you allow
axioms
to be stated in second-order logic for example, then itÕs
quite easy
to assert such things as R consists of ALL reals.
| Work at \finding the valid deductions in SOL
|or IF-logic if you care about set theory, use formal
deduction if you
|care about geometry. How does this relate to anything we are
talking
|about?
IsnÕt it about time for you to concede that your original
concern
as stated in the subject line-- SkolemÕs theorem-- is a
totally
lost cause? Now that no supporting argument relates to it.
[...]
|The PROBLEM is using ONE word set when people ARBITRARILY
CHOOSE to
|have the word essentially MEAN different things, itÕs bad
bad bad.
On what occasions to people decide to mean different things
by set?
Feel free to concede at any time that the notion of set is
univocal,
and consequently when we talk about real numbers, we are
talking about
just one system, which has uncountably many elements in it.
IÕm ready
to let you concede the failure of your original argument at
any time!
| If
|the word electron meant something different depending on who
said
|it, physics would get nowhere fast.
But there are several theories of it. If you agree that the
same term,
electron, can mean essentially the same thing, even though
there are
several theories of it, then you have to concede that the
term set
can have essentially the same meaning, although there are
different
theories involving it.
I think hardly anyone would claim that ZFC is supposed to be
a theory
about all sets. It naturally applies to the pure,
well-founded ones.
But people would generally prefer to single out these, since
they are
relatively easy to theorize about, than try to encompass
everything
in physics.
| I agree that someone could write
|out a proof (so could a Turing machine) without believing
the theorem
|to be true (so could a Turing machine), but if we are going
to use the
|word set it should mean something, not whatever the speaker
imagines
|in his head, thatÕs one of my biggest problems, \
IÕm trying
to get a
|straight answer about what is and is not a set, and IÕm not
getting
|one.
Maneuvering for that martyr status again, huh?
I donÕt recall you asking before, what is a set. The problem
is that
itÕs a suf\ficiently fundamental term that one \
canÕt really
provide a
de\finition of it, except in terms of other concepts that are
pretty
close relatives to it. ItÕs like asking in Euclidean
geometry, what
is a point. I gave you an explanation in terms of graph
theory, but
the concept of graph is very close to the concept of set.
This is not so unlike the situation in physical science. The
things
that are postulated in a theory need not have de\finitions. \
ItÕs
enough that the theory have observable consequences.
|> |Hintikka gives a justi\fication about why mathematicians
like the axiom
|> |of choice because it translates the standard
interpretation second
|> |order formulas into equivalent \first order formulas, but
then he shows
|> |that that doesnÕt work in general.
|> Mathematicians do tend to like instances of
quanti\fier-elimination,
|> even when they are unaware of the concept of
quanti\fier-elimination.
|
|IÕm not sure how thatÕs a response to my \
statement,
Because youÕre assuming that itÕs meant as an \
antagonistic
response!
I was simply agreeing that Mathematicians do, indeed, tend to
like
theorems that involve quanti\fier elimination. \
IÕm not sure
whether
that explains why they like the axiom of choice, but I
suppose it
could partially explain it.
Keith Ramsay
===
Subject: Re: SkolemÕs Paradox and why is math the way it is?
> J.E.:
> [...]
> |> In any case, those of us who are not formalists seldom
care whether
> |> the Riemann hypothesis is a theorem of ZFC or PA or
whatever, or
> |> whether the twin prime conjecture is a theorem of ZFC.
We do care
> |> about whether theyÕre true, however. The formalist
thinks somehow
> |> that these questions are not well enough de\fined, but
everybody
> |> else aside from Essenin Volpin as far as I know
disagrees.
> |You totally lost me here, people care about whether a
statement is
> |true when itÕs true in some models and false in others?
Just
> |consider the models where itÕs true, now \
itÕs true. Or
consider the
> |ones where itÕs false, now itÕs false. Why \
would anyone
care about
> |this? Am I therefore a formalist to \find this silly?
> You have to be something like a formalist, yes. But you
seem not to
> be very pure in your formalism. A pure formalist doesnÕt
think of it
> in terms of models, because a model is, after all, a set
(or a
class),
> which would imply that their actual foundation is some kind
of set
> theory. But a pure formalist doesnÕt start from set \
theory.
And then
> you start talking about IF-logic, which does not have a
formal
> consequence relation at all.
I tried learning what the term model meant to Skolem to
understand
his result (theorem? meta-theorem? I still donÕt know!) and \
I
didnÕt
understand the responses, most of apparant explantions seemed
like
attacks, many straw-man based. So IÕve given up on learning
SkolemÕs
result from Usenet and I have some books, a few by Quine and A
shorter model theory, so eventually IÕll \figure \
it out, but
until
then that topic is probably dead for now.
As for the term model when I say truth in all models that is
an
entirely different term. When Hintikka says model theory he is
quite honest in his book that he means logical semantics. I
have
another post where I discribe a boot-straping way to get
something
that IÕd use as a model and itÕs at the ame \
informal level
where
we de\fine what is a game or what is a formula, in fact
elements of
the universe of discourse should be formulas, the only
necessity of
an element of discourse is that you agree which team wins the
game
aeb for the elements a and b and that both sides agree the a
is
an element at the time when one side selects it. Formulas can
do
that, because for a \fixed N and a \fixed mechanical \
proof method
(MPM) you can make a model where you have as elements certain
oFOL
formulas T where N or T is not only a validity, but where it
is
provably so for the MPM. Then you have a domain of discourse,
so all
you need to have a model, is a way for the players to agree
on aeb
given a and b. If we donÕt care what the model is a model
of,
you can just say that the E-team wins all aeb, thatÕs a
model, no
recourse to sets. More interesting models exist, I discribed
one I
thought was interesting in my last post.
> I suspect many troubles have been created by the belief
that the
> point of view you are describing here is some kind of
consensus
> view in mathematics. Far from it! Quite a lot of
mathematicians
> would consider the question of whether there are models
relative
> to which a sentence is true of interest *only* because of
the
> possible light it might shed on whether they are true with
the
> terms meaning what they were intended to mean.
A Mathematicians can have interest for any reason they like,
but
they shouldnÕt expect others to EVEN KNOW what they are
talking about,
as in this case. What IS this standard meaning?
> DonÕt lose yourself in broad generalities here. Think of
the particular
> mathematical questions that we have, like the twin prime
conjecture.
> The twin prime pairs, 3,5; 5,7; 11,13; 17,19; 29,31; ...
appear to
> continue inde\finitely. Does it really seem plausible to you
that the
> answer to whether they really do could possibly be yes or
no, take
> your pick, depends on which axioms you use?
Yeah, it does, because different axioms will de\fine the \
natural
numbers differently. For instance if you had the single axiom
Axiom
of stupidity: Ax Ay xey & ~xey, then all the statements of
number
theory are true with the intended meaning of the axiom of
stupidity,
because the axiom implies all things vacuously. The axioms DO
matter.
> [...]
> |> I just went over to my bookshelf and opened a group
theory
> |> textbook at a random page. The theorem there was that
the center
> |> of the group GL(n,F) consists of the set of diagonal
matrices.
> |> GL(n,F) consists of the invertible n by n matrices with
entries
> |> in the \field F.
> |>
> |> What are the axioms that supposedly de\fine GL(n,F)? We
all know
> |> what natural numbers are, and what invertible n by n
matricies
> |> are, but not because there are axioms for them. An n by n
> |> matrix is a function from {1,2,...,n}x{1,2,...,n} to F;
invertibility
> |> means that there exists another such one that is its
inverse, etc.
> |>
> |> The starting point is arithmetic, i.e., knowing what it
means
> |> to have a natural number n. What complete axiomatization
of
> |> arithmetic do you have in mind when doing group theory?
> |IÕm really confused, I took this course called abstract
algebra and
> |we didnÕt assume any axiomatization of arithmetic, if \
fact
the whole
> |point was to avoid that, but instead to axiomize a group
so that
> |later anything that was a group would have the group
theory theorems
> |true of it.
> Certainly group is \first-order axiomatizable, but GL(n,F) \
has
> no \first-order axiomatization.
GL(n,F) is de\fined as a particular group, not as an abstract
group,
itÕs nature depends on the nature of the underlying set
theory.
> | The group GL(n,F) satis\fies the group axiom with the
> |multiplication that you would expect for it (any in fact
since F
> |satis\fies the \field axioms by hypothesis, you \
can prove that
GL(n,F)
> |is a group, which is GOOD because that makes the nomen
group
> |well-de\finedish with the fact that GL(n,F) satisifes the
group axioms,
> |the whole point is that the results we proved about the
center of
> |abstract groups can then be applied to the subcase of
groups GL(n,F).
> |Why you think this starts with arithmetic is comletely
behind me, you
> |have an abstract \field F, and from it you make a group
GL(n,F), where
> |does arthemetic come in?
> How did you de\fine GL(n,F) for an arbitrary natural number
n, if
> you didnÕt have any concept of natural number? What do you
think
> arithmetic is about, if itÕs not about natural numbers?
A natural number not an elementary term (like set or belongs
to)
and is usually DEFINED (in terms of set theory) as a set that
belongs to every set I in the intended model of set theory
such
that Ex (Ay ~yex) & xeI & (An neI => nU{n}eI). Or,
equivalently (for
an intended model) m is a natural number iff AJ (Ex (Ay ~yex)
& xeJ
& (An neJ => nU{n}eJ))=>meJ. This is well de\fined in a model.
Such
a number exists in a model with separation, union, pairing,
and the
right kind of in\finity, I think they might also exist just \
with
replacement & in\finity, not sure.
> In your abstract algebra course, did you talk about torsion
in
> an abelian group? In an abelian group, the set of elements
g such
> that n*g=e for some n, i.e. g+g+...+g (with n gÕs) is the
identity,
> are called the torsion elements, and they form a subgroup
of the
> group. But you have to be able to refer to natural numbers
n (or
> something that serves a similar purpose) in order to be
able to
> de\fine torsion.
> Group theory is full of results that are about speci\fic
kinds of
> groups, where the kind of group is not given by a set of
axioms.
The speci\fic groups are about speci\fic sets, their \
existance
(and
nature) is a piggy-back on set theory and depends on the
existance and
nature of the underlying sets.
> |> | \field theory,
> |>
> |> I donÕt really have a book just on \field \
theory as far as
I know,
> |> but it occurs to me that one \field being algebraic over
another
> |> isnÕt \first-order de\finable. \
The property of a \field
extension,
> |> that the over\field is a *\finite* dimensional \
vector space
over
> |> the sub\field, comes up often. The \finiteness \
intended is
what we
> |> (foolishly?) understood as just plain \finiteness, not
\finiteness
> |> relative to a model of [something].
> |The common axiomization of \field include the term set,
hence the
> |importance of the question what is a set.
> A group is also usually de\fined as a set with some \
additional
> structure. This is not a respect in which the de\finition of
> \field is different from the de\finition of \
group.
I didnÕt say it was, except aparantly down below it turns \
out
that
\field DOES have a SOL version because SOMEONE claims that the
axioms of a complete ordered \field are categorical.
> | If you threw in F is a
> |\field iff (FA1, and FA2, ... FAn, and STA1, STA2, ... , and
STAn)
> |(where FAk is a \field axiom and STAk is a set theory
axioms) then
> |youÕd know what the models of \fields are, but \
without, you
have to beg
> |the question over to set theory and ask is this a set to
know if
> |something is a model of the \field axioms.
> So are you beginning to see why I donÕt buy this idea that
> everything else works so wonderfully deductively until mean
olÕ
> set theory comes along and screws everything up? All these
lovely
> algebraic topics rely a little bit upon set theory. We
would be
> equally unhappy with the claim that a GL(n,F)-object in a
model
> of some theory is just as good as a real GL(n,F), as we are
with
> the idea that a real-line object in a model of some theory
is just
> as good as the real line.
Group theory and Field theory were de\fined in terms of sets in
the
courses I took, so itÕs NOT like they were \fine \
until set
theory came
along set theory was tehre to begin with and every thorn in
set
theory is a thorn in group theory, annoying me endlessly. If
there
are non-standard set theories, and with non-standard
natuarals, then
of course there are non-standard groups GL(n,F). The only
reason to
not like it is if you donÕt like the non-standard set \
theory.
And
if thatÕs the case just explain what the correct one is and
be done
with it.
> |> | geometry, etc.
> |>
> |> How many colors are needed to color the points inside a
unit
> |> square, so that no two points of the same color are a
distance
> |> of 1/2 apart?
> |>
> |> When people doing Euclidean geometry talk about the
Euclidean
> |> plane, they are talking about the one thatÕs isometric
to R^2,
> |> pairs of real numbers, not an arbitrary model of some
\first-order
> |> axiomatization of it.
> |Classical geometry is both complete (every statement with
the
> |uninterpreted primatives or geometry is either a theorem
or the
> |negation of a theorem) and categorical (all models are
isomorphic)
> The theory of elementary geometry, the \first order theory
having
> primitives for things like collinearity, congruence, and so
on, is
> indeed complete, but I intentionally chose a question about
Euclidean
> geometry that doesnÕt \fit in it.
> It is not categorical. No \first-order theory that has an
in\finite model
> is categorical.
This is so frustrating! I was told that the axioms of a
complete
ordered \field were categorical, but now youÕre \
saying that the
actual categorical is NOT the axioms we used in \field theory
(about
sets), but instead about SOL? And geometry too? How am I
supposed to
tell which people are lying to me?
> |as
> |an axiomatic theory, so what is arbitrary about itÕs
models? And
> |geometry doesnÕt have a primative color. \
YouÕve lost me
completely
> |again. IsnÕt that really a question about functions from
manifolds
> |into integers?
> Manifolds? One extremely speci\fic one.
You answered that above, I was discussing elementary
geometry, you
werenÕt. I think the confusion about that is now resolved.
> ItÕs commonplace in combinatorics to describe a function
from a
> domain to an arbitrary (usually \finite) set with a given
number
> of elements as a coloring. Questions about colorings of the
> Euclidean plane would ordinarily be categorized as
questions in
> Euclidean geometry, merely not elementary Euclidean
geometry.
You say something is common. OK. But I think later we get
back on
track.
> |> Generally, your statement comes much closer to correct
if we
> |> are considering relationships between second-order
statements.
> |> But thereÕs no formal deductive system for second-order
statements
> |> that captures all the valid deductions that can be made
in
> |> second-order logic. Second-order logic also involves
referring
> |> to arbitrary subsets of the domain, which is the usual
bugaboo
> |> of set theory.
> |Formal deductive methods donÕt work for set theory
correct, but they
> |do for geometry I donÕt know why you think they \
donÕt.
> I think I misunderstood the point of what you were saying
before.
> I was giving examples of theorems in these various areas of
> mathematics that arenÕt of the form, any structure
satisfying
> these axioms satis\fies this theorem too. You appeared to be
saying
> that set theory alone failed to \fit into this mold.
> There is little difference for the purposes of this
discussion
> between group theory and set theory. You indicated that
something
> seemed to stop working when it came to set theory, and I
think I
> misunderstood.
Elementary geometry is considered to be about the truths of
all model
for which the axioms of elementary geometry are not false. I
thought
all abstract math was like that, but if you have a model of
set
theory, then SOME people say to me things like thatÕs not
REAL set
theory, you are an idiot and shouldnÕt study set theory,
there is an
intended model unlike in elementary geometry and only that is
the real
set theory, this is surprising and new to me, never in any
class has
this happened before, this seemed to only happen with set
theory.
> There is no complete formal system for Euclidean geometry,
just
> for the so-called elementary part of it.
ThatÕs \fine, if you start doing set theory with \
some other
theory,
then it stops being just that other theory and becomes a
little part
of set theory, and then ... what? Formal systems stop and
informal
interpretation leap out at you? Seriously, what happens then?
Why is
there any change at all?
> | I donÕt know
> |why you keep bringing up formal deduction anyway, we \
donÕt
use
> |deduction to design theories, so what is this prima facie
reason for
> |such a hubbub about it?
> Hubbub? Oh, creating a hubbub, all by myself? I didnÕt
realize
> I had it in me.
You werenÕt the only one on this thread to \
\fixate on formal
deduction.
> Formal deduction is relevant because of its relationship
with models.
> (Essentially) the only thing that other models of ZFC have
in common
> with the cumulative hierarchy is that they satisfy the
*formally
> deducible consequences* of the axioms of ZFC, as
relativized to them.
Hmm, so are you saying that GoedelÕs theorem \
isnÕt true in
any model
except the intended one, do you have support for that claim?
> If you do not require the theory to be formal, if you allow
axioms
> to be stated in second-order logic for example, then itÕs
quite easy
> to assert such things as R consists of ALL reals.
SOL is informal now? SOL brings in SO assumptions which are
basically equivalnet to \first assuming sets exist, so it just
buries
the question what is a set into a circular trap. How is this
accomplishingly anything. You are just saying: assuming we
have all
the sets and nothing but the sets then we have all the reals
since the
reals are certain sets, I think I could live with that if you
told me
the \first part we have all the sets and nothing but the sets \
in
terms of formal SO axioms whose negation were translateable
into
IF-FOL.
> | Work at \finding the valid deductions in SOL
> |or IF-logic if you care about set theory, use formal
deduction if you
> |care about geometry. How does this relate to anything we
are talking
> |about?
> IsnÕt it about time for you to concede that your original
concern
> as stated in the subject line-- SkolemÕs theorem-- is a
totally
> lost cause? Now that no supporting argument relates to it.
I agree that Usenet was not successful to explain how
SkolemÕs results
jibe with the intended interpretation of set theory because I
still
donÕt understand SkolemÕs result well enough \
and Usenet
discussion is
not apparantly going to be able to work with that failing or
remedy
that failing. But I also asked why is math the way it is and
if
someone said because these are the SO set theory axioms
because I
said they were and enough people agreed that axioms were the
set
theory axioms then I think I could accept that. I canÕt make \
a
promise, IÕd want to actually SEE the axioms.
> [...]
> |The PROBLEM is using ONE word set when people ARBITRARILY
CHOOSE to
> |have the word essentially MEAN different things, itÕs bad
bad bad.
> On what occasions to people decide to mean different things
by set?
For instance that NIGHTMARE of a functional analysis class
where we
discussed all sets and changed half-way through the course
whether
the choice functions were sets or not!
> Feel free to concede at any time that the notion of set is
univocal,
> and consequently when we talk about real numbers, we are
talking about
> just one system, which has uncountably many elements in it.
IÕm ready
> to let you concede the failure of your original argument at
any time!
YouÕve claimed that the SO theory is categorical, but I \
donÕt
know
FO versions of real that I was taught. So I might be able to
concede in the future, but it is not yet possible.
> | If
> |the word electron meant something different depending on
who said
> |it, physics would get nowhere fast.
> But there are several theories of it. If you agree that the
same term,
> electron, can mean essentially the same thing, even though
there are
> several theories of it, then you have to concede that the
term set
> can have essentially the same meaning, although there are
different
> theories involving it.
These are COMEPTING theories, no one walks around and claims
that
these contradictory useages are ALL TRUE in some platonic
sense, and
de\finately not all true in reality.
> I think hardly anyone would claim that ZFC is supposed to
be a theory
> about all sets. It naturally applies to the pure,
well-founded ones.
> But people would generally prefer to single out these,
since they are
> relatively easy to theorize about, than try to encompass
everything
> in physics.
Then HOW do you know that there are MORE pure well-founded
sets that
are reals than pure well-founded sets that are naturals.
CanÕt you
just have some unpure or un well-founded set that acts as a
bijection between the two? The claim that the lack of
bijection IN
THE MODEL means something ABOUT THE SETS relies on the claims
that you
HAVE the be-all theory of bijections in your model, whether
that model
is all pure well-founded sets or some other model.
> | I agree that someone could write
> |out a proof (so could a Turing machine) without believing
the theorem
> |to be true (so could a Turing machine), but if we are
going to use the
> |word set it should mean something, not whatever the
speaker imagines
> |in his head, thatÕs one of my biggest problems, \
IÕm trying
to get a
> |straight answer about what is and is not a set, and IÕm
not getting
> |one.
> Maneuvering for that martyr status again, huh?
I would HATE to be a martyr, IÕm MUCH prefer to have people
explain
what they are saying so that I can understand what they are
saying.
> I donÕt recall you asking before, what is a set. The
problem is that
> itÕs a suf\ficiently fundamental term that one \
canÕt really
provide a
> de\finition of it, except in terms of other concepts that are
pretty
> close relatives to it. ItÕs like asking in Euclidean
geometry, what
> is a point. I gave you an explanation in terms of graph
theory, but
> the concept of graph is very close to the concept of set.
You are turning my arguments inside out, elementary geometry
DOES NOT
have an intended interpretation, ANYTHING that works is
considered
equally good, when I studied math, I was told AGAIN AND AGAIN
that
that was the POINT of PURE MATH, to have axioms WITHOUT
intended
meaning of the primatives. ItÕs called propositional
functions you
state something where are the non-logical terms can have
anything
stuck in them, as long as itÕs substituted fairly and
according to the
rules. With game theorectical semantics it means that you can
play a
game with any model, and a validity N or T is a validity,
full stop.
I was raised (mathematically) to assume that there IS NOT an
intended
interpretation of axioms and that doing so defeats the
purposed of
pure math. Now it appears that set theory is applied math,
which
is very confusing and counter-intuitive to me.
> This is not so unlike the situation in physical science.
The things
> that are postulated in a theory need not have de\finitions.
ItÕs
> enough that the theory have observable consequences.
If you donÕt have a de\finition, then \
shouldnÕt anyone me able
to apply
the theorems to anything that they consistently apply the
axioms to?
So if I have a model where some of the provable formulas are
the
domain of discourse, then canÕt I discuss how the statement
Ea (a is
a cardinal) & (|N| < a < |P(N)|) is true can be interpreted
as saying
that more statements can be added to the domain of discourse?
Why is
this sacriligeous? Of COURSE this domain of discourse is
countable
at the informal level if everything in the domain of
discourse is a
formula, but there isnÕt any statement IN THE MODEL that \
says
that
there is a statement IN THE MODEL that is a bijection IN THE
MODEL
between the statement that is the naturals IN THE MODEL and
the
statement that is the reals IN THE MODEL.
Being countable at the informal level (discussing the domain
of
discourse, countable-1), and being countable at the formal
level
(discussing properties of MEMBERS of the domain of discourse,
countable-2) are clearly different, so why is a countable-1
model
considered bad when there isnÕt any proof that a better \
model
exists,
and in fact I donÕt know how to boot-strap into a model that
is not
countable-1. I naively imagine that you have to assume sets
really
exist to get a model that isnÕt countable-1. The irony is
that the
diagonal arguement works at the informal level in such a way
as to
make me reject the naturals numbers existing at the informal
level.
But IÕd probably prefer to have someone show me a different
assumption
that I could negate other than the informal existance of the
countably-1 in\finite. LetÕs make a predicates \
out of a bunch of
predicates Pn(x), the new predicate says P(n) iff ~Pn(n) if n
is an
informal non-negative integer, and P(x) iff P0(x) otherwise.
So
having the informal notion of a natural number leads to
problems with
considering the truth of \finite statements. I need an informal
level
where I can discuss the truth of arbitrary \finite statements
MUCH more
than I need informal integers, so I reject the integers as
contradictory. So forgive me if IÕm skeptical when a formal
system
resurrects the integers, I worry about inconsistencies
reappearing,
and frankly I just assume that I am misunderstanding
absolutely
everything because surely this would bother other people if
itÕs not
simply the case that I screwed up big time.
> |> |Hintikka gives a justi\fication about why mathematicians
like the
axiom
> |> |of choice because it translates the standard
interpretation second
> |> |order formulas into equivalent \first order formulas, but
then he
shows
> |> |that that doesnÕt work in general.
> |>
> |> Mathematicians do tend to like instances of
quanti\fier-elimination,
> |> even when they are unaware of the concept of
quanti\fier-elimination.
> |IÕm not sure how thatÕs a response to my \
statement,
> Because youÕre assuming that itÕs meant as \
an antagonistic
response!
> I was simply agreeing that Mathematicians do, indeed, tend
to like
> theorems that involve quanti\fier elimination. \
IÕm not sure
whether
> that explains why they like the axiom of choice, but I
suppose it
> could partially explain it.
> Keith Ramsay
I alreadyy stated that I thought mathematicians wanted to
take the
STANDARD interpretation of the SO axioms AS WELL as the
standard
interpretation of the FO axioms, and that THAT was why they
liked
choice, but that that program fails because there is no model
that is
faithful to both, so why have choice?
BTW I wasnÕt expecting an antagonistic response, in fact I
strongly
appreciate that you are very kind in your responses, it just
seemed
like a non-sequitor and I didnÕt know if that meant that I
misunderstood it, thus my claim IÕm not sure how \
thatÕs a
response to
my statement and your answer that you werenÕt sure whether
that
explains why they like the axiom of choice, explained to me
that even
you werenÕt sure if that was an explantion so now I
understand that
you were proposing another reason. My reason was related to
the
conept of standard interpretations so I thought it related to
the
subject of the rest of the discussion, does that make sense?
===
Subject: Re: SkolemÕs Paradox and why is math the way it is?
|As for the term model when I say truth in all models that is
an
|entirely different term. When Hintikka says model theory he
is
|quite honest in his book that he means logical semantics. I
have
|another post where I discribe a boot-straping way to get
something
|that IÕd use as a model and itÕs at the ame \
informal level
where
|we de\fine what is a game or what is a formula, in fact
elements of
|the universe of discourse should be formulas, the only
necessity of
|an element of discourse is that you agree which team wins
the game
|aeb for the elements a and b and that both sides agree the a
is
|an element at the time when one side selects it.
One of the most disconcerting things about the whole
discussion
is that the term set in the broadest sense means model of the
\first-order predicate calculus with no non-logical axioms (or
the
empty set) and graph means model of FOPL with a single
binary relation. IÕll think some more about how you plan to
boot-strap
your way into accepting models \first, but at this point I
frankly do
not understand why accepting models as a starting point is
better
than accepting sets (or at least the sets in the cumulative
hierarchy)
as a starting point.
The sets in the cumulative hierarchy are always sets of some
particular kind of thing. Now, to avoid problems, we want our
kinds of things to be such as guarantees that any chain of
membership is grounded somewhere. If we allow it to end in
a non-set atom, then we have set theory with atoms. If we only
allow it to end in the empty set, then we have pure sets.
Keith Ramsay
===
Subject: Re: SkolemÕs Paradox and why is math the way it is?
Originator: joshp@xoxy.net (joshp)
> Think of the particular mathematical questions that we
have, like the
> twin prime conjecture. The twin prime pairs, 3,5; 5,7;
11,13; 17,19;
> 29,31; ... appear to continue inde\finitely. Does it really
seem
> plausible to you that the answer to whether they really do
could
> possibly be yes or no, take your pick, depends on which
axioms you
> use?
If I understand correctly, Harvey Friedman has found some
innocent-looking arithmetic statements whose truth values
actually do
depend on which axioms you use.
Theorem 1 is provable in EFA (exponential function
arithmetic).
Propositions 2,3 is provably equivalent, over ACA, to the
consistency
of MAH = ZFC + {there exists an n-Mahlo cardinal}_n.
--
Josh Purinton
===
Subject: Re: SkolemÕs Paradox and why is math the way it is?
> If I understand correctly, Harvey Friedman has found some
> innocent-looking arithmetic statements whose truth values
actually do
> depend on which axioms you use.
YouÕve misunderstood. The truth values of these statements \
do
not
depend on what axioms you use, but whether or not you can
prove them
depends on what axioms you use. In this, they are no
different from
e.g. for all natural number m and n, m+n=n+m.
===
Subject: Re: SkolemÕs Paradox and why is math the way it is?
Originator: joshp@xoxy.net (joshp)
>YouÕve misunderstood. The truth values of these statements
do not
>depend on what axioms you use, but whether or not you can
prove them
>depends on what axioms you use.
--
Josh Purinton
===
Subject: Re: SkolemÕs Paradox and why is math the way it is?
[...]
|I think the best way to teach quantum mechanics is to assume
that the
|wave-function is real (exists), and that the equations
describe how it
|moves, and thatÕs it, in practise thatÕs all \
you need and
every
|interpretation takes that seriously to the extent that the
|interpretation takes anything seriously at all.
I mostly agree, but at least one of my physics professors in
college
considered treating the wave-function as real as wrong,
wrong, wrong.
It applies only to a statistical ensemble, not to just a
single
system. (!) I would feel at least some qualms about leaving
students
with the idea that thereÕs a consensus opinion among
physicists about
the reality of it.
If the wave-function is real, then are the Everett-Wheeler
parallel
universes real too? I donÕt have a big problem with the idea
that
they are, but I know a lot of people do, which forces them
either to
abandon thinking that the wave function is real, or to assume
that
it really undergoes a reduction process, the nature of which
is
obscure.
|> To be completely unbiased and transparent with regard to
such
|> possible qualms as doubting that the natural numbers
really exist
|> probably seems like too much of a distraction. So the
usual approach
|> is basically not to worry about it. The platonist and the
formalist
|> will tend to sound the same as they are developing a
theory, since
|> deducing consequences from some assumptions typically
sounds just
|> like you believe the assumptions to be actually true in
some
|> domain. And then one can divert discussion of qualms to
such venues
|> as sci.math.
|
|This seems rather ahistorical, the problem is that at one
point people
|took the existance of classes for granted and it created
problems.
Nearly every reference to a problem turns out on closer
inspection
to be a reference to Frege. Are you talking about someone
other than
containing a self-contradictory axiom, we should now all be
worried
about the safety of using set theory.
Most mathematics, incidentally, uses only a relatively
uncontroversial
portion of set theory. People deal with things like sets of
real
numbers all the time, but not so often the parts that depend
on
(say) the axiom of replacement.
|And the point of the modern theory is to avoid those
problems, so
|you have to be clear that everybody is doing the same thing
so that if
|someone gets a problem itÕs clear that the system is to
fault, not the
|person.
Well, Frege succeeded in this as well. He made his own
system, and
remarked that any deduction could be tracked back to the
axioms,
so that if there was a problem, it would be with one of the
axioms.
And then it actually occurred, with the problem lying with
axiom 5.
|> | We can assume induction in the langauge and then latter
|> |show there there is an induction INSIDE the theory as
well, so that we
|> |donÕt have to use induction outside the theory, but
thatÕs very very
|> |different than proving induction without proving
induction. ThatÕs
|> |proving induction in a theory using induction outside the
theory.
|> Yes, *if* you assume induction for your original concept
of string,
|> itÕs an informal assumption that canÕt come \
from inside
the theory.
|> Having proven mathematical induction from the axioms, to
conclude that
|> it applies to actual strings is a further step, requiring
either
|> believing the axioms are correct in some sense or
something like that.
|
|As I mentioned before, the PURPOSE of the formal theory was
to avoid
|problems, if you actually are depending on the informal
theory (IÕm
|reading Quine now and so far it looks like formulas will be
built out
|of philosophical statements and not strings and that
statements of
|set theory will be based on other statements, and that the
axioms will
|likely be about assuming the truth of some statements (which
is like
|interpreting a schemata), I havenÕt \finished it \
yet, so donÕt
think
|thatÕs how IÕm characterizing Quine, \
itÕs just my
expectation based on
|where I am so far and it seems at least not to be circular
at this
|point.
I donÕt know where your \first left parenthesis \
is supposed to
be closed.
The formal theory has various purposes, many of which could
be lumped
together under avoiding problems. But there are things that
it canÕt
do for you. It doesnÕt, by itself, tell you what the
statements of the
theory are supposed to mean. You can, if you like, punt on
that question
and just proceed to work with the theory formally. But if you
are going
to worry about the questions that depend on the actual
meaning of
statements, it all has to start out somewhere informally.
[...]
|Set theory was billed to me as the type-free be-all theory,
and IÕm
|not sure if you are refuting that as a misrepresentation
that my
|teachers made or if yoy are agreeing with them, I canÕt \
tell.
IÕm not sure what type-free be-all theory means. \
ItÕs type
free
in a sense. ItÕs not like the type theory of Martin-Lof or \
of
Principia Mathematica.
I donÕt think there is a be-all theory.
|But IF
|logic avoids having in\finite regresses into higher-order
logics so
|that we CAN sit down and discuss how you make theories, so
isnÕt that
|worth considering?
What in\finite regress into higher-order logic is there for
anyone?
| Consistent theories imply strategies. ThatÕs a
|REAL implication. But people complain against IF-logic
exactly
|because itÕs the right size to do that because it \
isnÕt the
right
|size for NOT doing that (not the right size for formal
deduction).
|And thatÕs silly because IF-logic has ordinary FOL as a \
part
of it
|anyway, so anything you do with oFOL you can do with
IF-logic, just
|stop using the / or // symbols.
I donÕt complain about IF-logic because itÕs \
not formal
(i.e., has
no complete set of deductive rules); I only fail to see why
once
one has abandoned formality one should pick IF-logic over
languages
that more directly talk about sets.
|> For theories with more realistic goals, I would say
self-application
|> is a bit like being able to lift all the rocks that one
can make.
|> It could be a sign that one is strong. Or it could be a
sign that one
|> has a limited ability to make rocks. IF logic can de\fine
its own
|> truth-predicate, yes. But thatÕs a combination of being
strong in some
|> ways, and being weak in others.
|
|In what way do you think it is weak?
You canÕt say that a graph has three connected components, \
in
it.
|> | but that doesnÕt mean there isnÕt a
|> |strong theory that *can* do itÕs own model theory that
has set theory
|> |(and hence everything based on it) as a component. \
ThatÕs
what IÕm
|> |looking for now, and I think the excluded middle is the
only thing in
|> |the way really. There is a subsection of the universe
where the
|> |excluded middle holds, and thatÕs what we call set
theory, but itÕs
|> |intended models (if it has any) live outside that
subsection.
|> Why?
|
|Every model of set theory lacks a set that should exist as
much as the
|alleged uncounted real should exist.
If by model you mean a set with an epsilon relation on it,
then
this is correct, but people often mean by model either a set
*or*
a proper class with an epsilon relation on it. The cumulative
hierarchy does not lack a set that should exist-- it consists
by de\finition in all the well-founded pure sets.
[...]
|And once you extend the de\finition of set to have \
non-excluded
|middles, the standard proof about the lack of a set with a
speci\fic
|property goes away, the theorem becomes the graph of the
bijection
|between a set and itÕs power set does not have an excluded
middle,
|even if it exists. And yes we can make a hierarchy based on
|equivalnce classes of sets based on graphs with excluded
middles,
|itÕll be just like cardinality theory if we do it right.
The proof that a set does not have a bijection with its own
power
set works \fine in intuitionist logic, which \
doesnÕt incorporate
the law of excluded middle as a rule of inference. If
f:X->P(X)
is a function, then {x in X : x is not in f(x)} is a set that
is
not in the image of f. If f(y) = {x in X : x is not in f(x)},
then
y is in f(y) if and only if y is not in f(y). That is
inconsistent
in intuitionist logic.
I think youÕre avoiding this conclusion by considering it
possible
for some statement to be true if and only if it is not true. I
understand how IF-logic permits such a thing to occur, but
itÕs
not convincing to me that this makes good sense.
Once we have a logic with three truth-values, true, false and
indeterminate, I donÕt see how it can be invalid for me to
start
talking about which of the three bins a sentence falls into.
The
claim that a certain sentence simply fails to be true, i.e.,
is
either false or indeterminate, appears to make logical sense.
ItÕs
just not a claim that can be expressed in IF-logic.
This is really what I want Hintikka to tell me: why am I
mistaken
when I think IÕve made an assertion such as the domain of
discourse
is \finite that is true *exactly* when a certain sentence of
IF-logic
fails to be true. In what way is this an incoherent sentence?
It
seems as though he simply ham-strings his theory to make it
impossible
to say certain things in it.
Wittgenstein gave an example of an incoherent description:
when itÕs
\five oÕclock on the Sun. He imagines someone \
asking, what does
that
mean? and the answer is, Just like what it means to be \five
oÕclock
here-- but on the Sun. I can imagine that some of the things
I think
and talk about are confused in sort of this way: it only
seems to me
that IÕm considering a well-de\fined proposition. \
The only way
I can
see how it would make sense to consider a system like
IF-logic to be
ultimate is if I were confused in such a manner about the
things that
I say that appear not to be expressible in IF-logic.
[...]
|I can grant that no deception was intended if you think it
was just a
|technical inaccuracy, IÕm a bit scarred (as in maimed, not
as in
|afraid) in that I still do not know the order in which to
resolve
|things, but IÕm still hoping this book of \
QuineÕs IÕm
reading will get
|everything in the right order.
It seems a bit unlikely that there was an intention to
deceive, although
those of us reading you on sci.math have no \first-hand
knowledge of it.
I think it will be mighty dif\ficult for you to order your
development
unless you admit the necessity of starting informally, for at
least a
brief time, or you decide to go formal all the way and not
care about
such things as whether the theory has a model.
Lorenzen likened the process to the process of building a
ship at sea.
With one ship, you can build another one inside of it. But if
you are
just tossed out into the ocean, with some building materials,
you have
no choice but to learn to swim, and then maybe build your
system while
swimming.
Figure out what original unde\fined terms you are willing to
accept.
Then decide what axioms about them you are willing to accept.
Then
go from there.
[...]
|I think this is
|about describing separation badly, I donÕt think \
itÕs about
logic or
|language at all, so is EF ~ (Ax Ey Az (zey) <-> (zex and
0=F[z])) a
|good descrpition of what is intended by separation or not? I
still
|donÕt know.
What is intended by separation being false, do you mean? If
IÕm not
misunderstanding your notation, this is what I remember as
being the
second-order separation axiom (negated).
[...]
|> DonÕt confuse rigor with formality. Only a formalist \
needs
to have
|> formula de\fined *inside* of a formal theory separately from
outside.
|
|IÕm unfamiliar with your de\finitions of \
realist, formalist
and so
|on, it just wasnÕt covered in my education. Some people on
this
|Usenet group have told me to take a set theory class, I
have, they
|didnÕt cover those terms. Are they covered inmost classes
and was I
|just unlucky?
I donÕt know of many places offering courses that would \
cover
it.
Philosophy of mathematics is not a big \field, and I \
wouldnÕt
be at
all surprised if you went somewhere where there simply
werenÕt any
courses in it. You could try something like the Benacerraf
and Putnam
book.
|IÕm \fine with IF-logic saying that some string \
represent
well-de\fined
|games and that some games have winning strategies for one
side, and
|some for the other, and some donÕt. IÕm \
\find with someone
making a
|claim that the set theoretical universe is such as to make
the string
|(~A1)or(~A2)or...(~An)or(T) true (when ßeshed out with the
right
|axioms for A1 through An and the theorem for T,
Actually, the chunk V_a of the set-theoretical universe I was
referring
is supposed to make these negated axioms ~Am fail to be true.
Of course
one canÕt say simply fail to be true in IF-logic. The string
(~A1)or(~A2)or...(~An)or(T) is supposed to hold true in _all_
domains
of discourse, since for all of the ones other than the
intended chunk
of the set-theoretical universe, one of the (~Am) holds. And
for that
intended one, (T) does.
| and if they say that
|itÕs true for all theorems, then that just leads to the
question what
|are the theorems, but that isnÕt confusing because we are
forever
|talking about actual games and these questions are about
what elements
|can be selected for substitution in the games and which
atomic
|sentances AeB are going to be true, and which are going to
be false.
|ItÕs forever a discussion about the rules of the game, no
in\finite
|regress into types.
It seems, then, that we have found a dictionary between
whatever I
might have to say mathematically that concerns this large
submodel
of the cumulative hierarchy (the sets having rank less than
the
smallest inaccessible cardinal) can be translated into a
language
that you can understand!
|This is FINE for physics because in physics we ALSO play
veri\fication
|and falsi\fication games in the laboratory, so I can make them
match
But you donÕt play uncomputable strategies against the
universe,
so far as I know. IF-logic only works the way that it does
because
one is implicitly assuming the possibility of using
uncomputable
strategies.
How else can it be true that either for every x, ~P(x), or
there
exists an x such that P(x)? The only way to win that game is
to
be able to search the whole domain of discourse to check
whether
any of the elements satis\fies P!
|The
|universe is such that when I do this experiment I get this
kind of
|results, and one can make DIFFERENT models to help CHOOSE
new things
|to TEST (in both math and physics). And based on the
results, you
|might decide to change the rules of the game (new axioms),
or just
|make new de\finitions to make existing questions easier (for
people) to
|ask, verify, or falsify.
This story holds equally well regardless of what axioms we
choose
to use, however. In fact, if all you care about is generating
testable predictions, you should refrain from worrying about
such
things as whether the mathematical axioms have a model of the
kind
people think they do. Just deduce consequences! Seriously.
[...]
|The physics classes IÕve taken I can teach myself, why is
math so into
|hiding things? I couldnÕt honestly teach set theory today,
even a
|basic one, because I havenÕt seen a logical presentation. \
My
physics
|teachers would answer questions when the students got
together and
|demanded resolution (like when we asked to know how you know
when to
|treat a stick as single object versus each molecule like a
separate
|object, versus each atom as an object versus electrons and
nucleuses
|versus electons and quarks and gluons), and they did so in
|non-circular ways.
Almost nothing youÕve asked about has been about the logical
development of the subject. (Do you have any questions of the
form, Is X a theorem of Y? or Is X an axiom of Y?) There
has been scarcely any strictly mathematical (as opposed to
philosophy of math) question in this discussion. You ask
things
like How do theorists know that the SEQUENCE to generate h
[PlanckÕs constant] exists in ZF? that doesnÕt \
have any clear
meaning.
I donÕt think physics classes and mathematics classes are
different
the way you think. Over and over, in physics we were taught
theories that we knew were not quite correct, but which we
were
supposed to value as useful approximations to the truth,
helpful
in seeking better approximations later. Our main job was to
work
with them, in spite of whatever imprecisions were involved.
There were a number of places where we had to fudge. \
In\finities
appear in certain places that one just has to accept as being
not quite right. The energy in the electric \field of a charged
canÕt be correct; the product of the charge and the electric
\field vector at it is unde\fined, because the \
electric \field goes
small and so on. The notation A << B is tossed around with gay
abandon, meaning usually that weÕre supposed to pretend that
(A/B)^2 might as well be 0. The business about the force law
is,
incidentally, not so trivial to correct.
What exactly is the Dirac delta function? Well, itÕs not
really
a function, and telling you what it precisely is, is beyond
the
scope of this course. :-) This willingness to work with an
ill-de\fined
concept is not just accepted; actual pride is taken by
physicists
in their willingness to dispense with precise de\finitions and
just work intuitively with stuff. One E&M instructor told us
that
a mathematician would say that f(x)=e^x does not have a
Fourier
transform, but that he would say it does have a Fourier
transform,
but we just donÕt understand what kind of thing it is! We
solved
the problem of a plane wave incident on a circular obstacle by
integrating a function that behaves like e^{i * (x^2+y^2)}
as x and y go to in\finity. (Where it oscillates rapidly,
pretend
Experimental sciences all depend on the arrow of time in a
way that
is not very often explained. I mention this little gotcha
partly
because to me it has a family resemblance the alleged
circularity
involved in using a formal system to prove results about
itself.
One is forced, ultimately, to make assumptions about the
initial
state of the universe, just as in mathematics one is forced,
ultimately,
to assume something about how it applies to the world.
Little bits of slop everywhere. Please note that IÕm not
complaining
about the courses I took or the instructors I had. I think
they
did a \fine job overall, and they were right to expect us to \
put
up with some of the messy bits of the subject, basically not
to
get too hung up on the parts that have to be fudged. We were
expected to keep going in spite of it.
The difference in this respect is the in mathematics, we come
ever
so much closer to getting it exactly correct, and ironically
get
to suffer as a result. People come to expect a much higher
degree
of exactitude. If you were not traumatized by the inability
of your
physics courses to explain *precisely* how we are supposed to
get
away with all this stuff, but were traumatized by your
mathematics
doing so-- the only explanation I can \find is that you
approached
the two with very different expectations.
Now, the one thing youÕve suggested is that at least one of
your
mathematics professors failed to acknowledge the imprecision
in
his remark about the axiom of comprehension. I would have to
hear
just exactly how that exchange went between the two of you
before
I would assume that this is so. If he did pretend to be
absolutely
precise when he was not, however, IÕm sorry that he did.
I wouldnÕt say that itÕs much worse than the \
physics
professor who
once was diagonalizing a matrix in class, and did it wrong. I
pointed
out his mistake, and he muttered that I couldnÕt have \
\figured
it out
like that, erased his work, and then did it over more
carefully.
|But the math proffessors just say why are you so
|interested in foundations, I thought you liked physics?, \
IÕm
just
|simply tired of being discriminated with, IÕm certain that
they talk
|not in circles amongst themselves,
IÕve never had a lot of conversations with set theorists, \
but
I can
assure you, conversations between professors of mathematics
are no
more formal, rigorous, and ultra-precise than their
conversations
with students. They are much less liable to be tripped up by
inadvertent imprecisions of the kind youÕve been complaining
about,
and consequently they take less care to avoid them amongst
themselves.
The kind of stuff youÕve been agonizing about is just not
worried
about! When writing papers, of course, thereÕs a third
standard that
is in some ways more formal, but also allows for details that
the
(professional) reader doesnÕt need to be left out.
|and I think itÕs down right rude to
|hide the actual logical developement from people just for
not being
|in the club, this isnÕt middle school this is science.
Perhaps you think that at some later date, the students who
mean
to go on to become professional set theorists get taken aside
and
have the real development of the subject taught to them. I
donÕt
think so.
| I consider
|math to be science.
Then try treating it like one.
The key objections to be made to a physical theory are that
its
predictions are observed to be incorrect, itÕs been
superceded by
a more comprehensive theory, and that itÕs needlessly \
complex.
The corresponding objections can be made of a mathematical
theory
too, in principle, but I see you making an awful lot of
objections
that are of a completely different kind.
Allow it to be just as sloppy and messy as physics is and
nearly
all of those remaining troubles are gone! If you are still of
the
impression that you canÕt tell what ZF is, try naming some
statement
which you either have trouble knowing how to formalize, or
donÕt know
whether itÕs an axiom, or a proof which you \
donÕt know
whether itÕs
valid, and we can surely clear it up for you.
[...]
|There is the translation from English to set theory and back
|constantly, there is no way for me to tell that IÕm doing \
it
the same
|way. The point is that someone can say X is a theorem: Y,
QED but
|if you donÕt know WHAT it is a thoerem of, then I \
canÕt turn
around to
|the guy next to me and say X is a theorem because if IÕm
asked
|theorem of what then I canÕt answer, so the theoremhood of
the
|statement isnÕt really proven (I canÕt carry \
it around with
me or
|apply it outside of the set theory class), it was only
stated as a
|theorem of SOMETHING. Only after we know the axioms can we
know that
|X is IN FACT a theorem of THAT axiom system.
We donÕt depend upon axiom systems in the way that you
imagine.
When someone claims that a result is a theorem, they mean
that it
has been proven, period. End of sentence. Not, proven in...
but
proven.
It is true that having proven a result, your proof can be
examined
to see what kinds of axioms would suf\fice for it. If you ask
what
itÕs a theorem in, you are liable to get the answer ZFC, not
because the proof has been examined in this way, but just
because
itÕs so rare for any other assumptions to be used. If one
does use
an assumption not supported by ZFC, itÕs expected that one
will
mention it somewhere in the course of the proof.
[...]
|> I had in mind the common situation where one has a
\first-order theory.
|> In that case, we have the Goedel completeness theorem that
says the
|> logical consequences of a set of axioms are the same as
the consequences
|> that can be deduced using standard \first-order logic.
|
|Are you sure about how you stated that? IÕm assuming \
standard
|\first-order logic is ordinary \first order logic \
(so not
IF-logic or
|SOL), but with IF-logic you can make \first order statements
that
|arenÕt statements of ordinary \first order \
logic, so you claim
seems,
|... a bit sensational. If itÕs true, then \
thatÕs great, but
I want to
|know if thatÕs what you meant.
Hintikka misleads the unwary reader by causing him (i.e. you)
to
think that \first order statement has come to mean something
more
than a statement in ordinary \first-order logic. It has done so
only in HintikkaÕs own terminology. He claims his system
should
count as \first-order logic, but this is not what anybody else
means
by it.
Certainly the Goedel completeness theorem is only for
\first-order
logic, not IF logic.
Keith Ramsay
===
Subject: Re: SkolemÕs Paradox and why is math the way it is?
>I mostly agree, but at least one of my physics professors in
college
>considered treating the wave-function as real as wrong,
wrong, wrong.
>It applies only to a statistical ensemble, not to just a
single
>system. (!)
DoesnÕt the Aspect experiment make that view untenable?
--
Shmuel (Seymour J.) Metz, SysProg and JOAT
<http://patriot.net/~shmuel>
Unsolicited bulk E-mail subject to legal action. I reserve the
right to publicly post or ridicule any abusive E-mail. Reply
to
domain Patriot dot net user shmuel+news to contact me. Do not
===
Subject: Re: SkolemÕs Paradox and why is math the way it is?
>>I mostly agree, but at least one of my physics professors
in college
>>considered treating the wave-function as real as wrong,
wrong, wrong.
>>It applies only to a statistical ensemble, not to just a
single
>>system. (!)
>DoesnÕt the Aspect experiment make that view untenable?
It also indicates a lack of understanding of probability, and
also quantum mechanics.
--
This address is for information only. I do not claim that
these views
are those of the Statistics Department or of Purdue
University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
===
Subject: Re: SkolemÕs Paradox and why is math the way it is?
>>I mostly agree, but at least one of my physics professors
in college
>>considered treating the wave-function as real as wrong,
wrong, wrong.
>>It applies only to a statistical ensemble, not to just a
single
>>system. (!)
>DoesnÕt the Aspect experiment make that view untenable?
> It also indicates a lack of understanding of probability,
and
> also quantum mechanics.
AFAICT, quantum theory only explains the statistical behavior
of
matter scienti\fically. Naturally, a statistical explanation
can refer
only to ensembles. That the wave function or fundamental
randomness is
real is a further philosophical interpretation which is not
part of
the science. (It seems...)
--
Eray Ozkural
===
Subject: Re: SkolemÕs Paradox and why is math the way it is?
> [...]
> |I think the best way to teach quantum mechanics is to
assume that the
> |wave-function is real (exists), and that the equations
describe how it
> |moves, and thatÕs it, in practise thatÕs \
all you need and
every
> |interpretation takes that seriously to the extent that the
> |interpretation takes anything seriously at all.
> I mostly agree, but at least one of my physics professors
in college
> considered treating the wave-function as real as wrong,
wrong, wrong.
> It applies only to a statistical ensemble, not to just a
single
> system. (!) I would feel at least some qualms about leaving
students
> with the idea that thereÕs a consensus opinion among
physicists about
> the reality of it.
There is a result that predates QM that says reproducable
correlation
implies some degree of systemic common cause. A certain
amount of
correlation could be chance, but if again and again and again
you see
it, then something has to cause the correlation. Without this
priciple there is no point to science because otherwise no
matter how
much you do something someone can say you were lucky.
OBSERVATIONS of the wave-function apply ONLY to ensemles. But
SOMETHING has to exist to be the common causes, and you can
demonstrate rather conclusively that anything else being the
common
cause can be disproven. IÕm \fine with telling \
students that
there is
no consensus view, but IÕd still present the common-cause
theorem, and
showing the students how it is inconsist to assume that QM is
correct
and that something other than the wave-function is the common
cause.
> If the wave-function is real, then are the Everett-Wheeler
parallel
> universes real too? I donÕt have a big problem with the
idea that
> they are, but I know a lot of people do, which forces them
either to
> abandon thinking that the wave function is real, or to
assume that
> it really undergoes a reduction process, the nature of
which is
> obscure.
THe wave-function exists in con\figuration space, which is
increadibly
large. To have two Everett-Wheeler universes interact with
each
other is on a par with all the air in your room appearing on
one side
of the room a la the statistical mechanics example of the
highely
unlikely. I take ufront to claims that there are parallel
universes
in the sense that they canÕt interact with each other \
because
if you
put the reverse Hamiltonian in, then it can, but thatÕs just
like if
you reverseed every molecule, then you could drive the
process of a
glass falling off a table and shattering backwards to turn
air and
friction and shards coluding to form a whole glass that has
the
momentum to jump up to the table and come to rest. ItÕs
possible, but
we arenÕt going to be able to do it in the lab, but we \
discuss
thermodynamics of observables and the fundamental micro-laws
separately becuase they are separate, I think Everett-Wheeler
is a bad
theory to mix the two fundamentally different concepts as if
they are
a single concept. They are two (1) the Quantum wave-function
evolves
according to the evolution operator (schrodinger
schrodingler-pauli,
dirac, etc. depending on the level of accuracy) and (2)
statistically
the physical state of some macrostates (incomplete
despriptions of
physical state whose macrostate appears equivalent to the
actual
> |> To be completely unbiased and transparent with regard to
such
> |> possible qualms as doubting that the natural numbers
really exist
> |> probably seems like too much of a distraction. So the
usual approach
> |> is basically not to worry about it. The platonist and
the formalist
> |> will tend to sound the same as they are developing a
theory, since
> |> deducing consequences from some assumptions typically
sounds just
> |> like you believe the assumptions to be actually true in
some
> |> domain. And then one can divert discussion of qualms to
such venues
> |> as sci.math.
> |This seems rather ahistorical, the problem is that at one
point people
> |took the existance of classes for granted and it created
problems.
> Nearly every reference to a problem turns out on closer
inspection
> to be a reference to Frege. Are you talking about someone
other than
> containing a self-contradictory axiom, we should now all be
worried
> about the safety of using set theory.
IÕll discuss this later in respose to Cantor.
> Most mathematics, incidentally, uses only a relatively
uncontroversial
> portion of set theory. People deal with things like sets of
real
> numbers all the time, but not so often the parts that
depend on
> (say) the axiom of replacement.
Since we havenÕt proven that ALL the set theory axioms taken
together
are consistent, then youÕd expect that if a smaller subset
works for
physics, that someone would have tried to prove that that
smaller set
of axioms was consistent. Is there such a proof?
> |And the point of the modern theory is to avoid those
problems, so
> |you have to be clear that everybody is doing the same
thing so that if
> |someone gets a problem itÕs clear that the system is to
fault, not the
> |person.
> Well, Frege succeeded in this as well. He made his own
system, and
> remarked that any deduction could be tracked back to the
axioms,
> so that if there was a problem, it would be with one of the
axioms.
> And then it actually occurred, with the problem lying with
axiom 5.
You missed my point, IÕm saying that if we eplain axioms
badly, then
people could interpret the axioms in two ways (one
contradictory, the
other not) and that if other things are de\fined in terms of
the axioms
then the de\finitions would be different, and then two people
could
have the case that one comes to a contradiction and the other
doesnÕt
and they could trace this DISagreement BACK to a disagreement
about
the interpretation of the axiom. If the author were dead by
that
point, then we wouldnÕt know what the correct theory was, \
and
in
fact weÕd have to start the process that the author should
have done
in the \first place which is to rewrite the axioms so that all
the
interpretations are equivalent in what the theory predicts.
> |> | We can assume induction in the langauge and then latter
> |> |show there there is an induction INSIDE the theory as
well, so that
we
> |> |donÕt have to use induction outside the theory, but
thatÕs very very
> |> |different than proving induction without proving
induction. ThatÕs
> |> |proving induction in a theory using induction outside
the theory.
> |>
> |> Yes, *if* you assume induction for your original concept
of
string,
> |> itÕs an informal assumption that canÕt \
come from inside
the theory.
> |> Having proven mathematical induction from the axioms, to
conclude that
> |> it applies to actual strings is a further step,
requiring either
> |> believing the axioms are correct in some sense or
something like that.
> |As I mentioned before, the PURPOSE of the formal theory
was to avoid
> |problems, if you actually are depending on the informal
theory (IÕm
> |reading Quine now and so far it looks like formulas will
be built out
> |of philosophical statements and not strings and that
statements of
> |set theory will be based on other statements, and that the
axioms will
> |likely be about assuming the truth of some statements
(which is like
> |interpreting a schemata), I havenÕt \finished \
it yet, so
donÕt think
> |thatÕs how IÕm characterizing Quine, \
itÕs just my
expectation based on
> |where I am so far and it seems at least not to be circular
at this
> |point.
> I donÕt know where your \first left parenthesis \
is supposed
to be closed.
IÕm thinking that at the current end of the paragraph and
that the
rest of the paragraph was accidentally killed, sorry. IÕm \
not
even
sure anymore what induction means, I was reading about the
foundation
axiom and I get e-induction, but the standard induction
depends on a
de\finition of natural numbers or something, which \
canÕt be both
outside the theory and inside the theory at the same time,
they have
to be different inductions. If itÕs neccissary outside the
thoery,
then I donÕt know why it isnÕt put in as an \
axiom inside the
theory,
since the old one can be brought inside.
> The formal theory has various purposes, many of which could
be lumped
> together under avoiding problems. But there are things that
it canÕt
> do for you. It doesnÕt, by itself, tell you what the
statements of the
> theory are supposed to mean. You can, if you like, punt on
that question
> and just proceed to work with the theory formally. But if
you are going
> to worry about the questions that depend on the actual
meaning of
> statements, it all has to start out somewhere informally.
With IF-logic, there is an informal level, itÕs about \
winning
strategies, I can handle that because I know what that means,
and that
assigns a meaning to every formula as being about a game
played in a
model (which is IMO just an agreement with both sides on what
the
universe of discourse is and about who wins a game of xey for
evey x
and y in the universe of discourse). Nothing more, and
nothing less.
When I was reading about the foundation axiom I saw a proof
that there
is no set of all games, I thought that was an interesting
fact.
> [...]
> |Set theory was billed to me as the type-free be-all
theory, and IÕm
> |not sure if you are refuting that as a misrepresentation
that my
> |teachers made or if yoy are agreeing with them, I canÕt
tell.
> IÕm not sure what type-free be-all theory means. \
ItÕs type
free
> in a sense. ItÕs not like the type theory of Martin-Lof or
of
> Principia Mathematica.
> I donÕt think there is a be-all theory.
IsnÕt being a be all theory part and parcel of the standard
interpretation that everything that could exist for anyone
that is
small enough to be a set, is a actually a set? The only
reason to
insist on this rather than just that enough sets exist to
satisfy the
axioms is because one wants to pretend like one can have an
everything, where the universe of discourse of standard
interpretation set theory is superior to all other universe of
discourses.
Otherwise why is that interpretation consider necissary or
even
standard?
> |But IF
> |logic avoids having in\finite regresses into higher-order
logics so
> |that we CAN sit down and discuss how you make theories, so
isnÕt that
> |worth considering?
> What in\finite regress into higher-order logic is there for
anyone?
Like you say in your previous post, you need SO set theory to
de\fine
the strongly inaccessible cardinal, in order to get a
faithful model
of set theory, but once you introduce SO set theory, people
will want
the other sets too, because the whole POINT of introducing
that
cardinal was to get all the sets that were missing in previous
models. You arenÕt succeeding at getting all the sets.
> | Consistent theories imply strategies. ThatÕs a
> |REAL implication. But people complain against IF-logic
exactly
> |because itÕs the right size to do that because it \
isnÕt
the right
> |size for NOT doing that (not the right size for formal
deduction).
> |And thatÕs silly because IF-logic has ordinary FOL as a
part of it
> |anyway, so anything you do with oFOL you can do with
IF-logic, just
> |stop using the / or // symbols.
> I donÕt complain about IF-logic because itÕs \
not formal
(i.e., has
> no complete set of deductive rules); I only fail to see why
once
> one has abandoned formality one should pick IF-logic over
languages
> that more directly talk about sets.
I wouldnÕt say I prefer to use IF-logic to talk directly
about sets,
in fact itÕs to have that informal level that I think I can
share in
common with others, I want a shared theory and I \find that I
have
dif\ficulty taking the standard interpretation of set theory
seriously
whereas I can take the existance of games (\finite games even!)
seriously. I donÕt know of a language that talks about sets.
If I
wanted to talk about sets IÕd choose some SO axioms,
translate their
negations into IF-logic, or them together to get N and then
IÕd not
that for every theorem N or T is a truth in all models, and
that is
a statement that I understand, that is falsi\fiable even if I
claim to
have a proof (nice for physics), and that doesnÕt even
require that I
assume N to be false in any model, in fact as long as it
isnÕt true in
a model M, then all the (FO) theorems
T are true in M.
> |> For theories with more realistic goals, I would say
self-application
> |> is a bit like being able to lift all the rocks that one
can make.
> |> It could be a sign that one is strong. Or it could be a
sign that one
> |> has a limited ability to make rocks. IF logic can de\fine
its own
> |> truth-predicate, yes. But thatÕs a combination of being
strong in some
> |> ways, and being weak in others.
> |In what way do you think it is weak?
> You canÕt say that a graph has three connected components,
in it.
Do you have a citation for that result, or better yet can you
state
your de\finition of graph and connected component?
> |> | but that doesnÕt mean there isnÕt a
> |> |strong theory that *can* do itÕs own model theory that
has set theory
> |> |(and hence everything based on it) as a component.
ThatÕs what IÕm
> |> |looking for now, and I think the excluded middle is the
only thing in
> |> |the way really. There is a subsection of the universe
where the
> |> |excluded middle holds, and thatÕs what we call set
theory, but itÕs
> |> |intended models (if it has any) live outside that
subsection.
> |>
> |> Why?
> |Every model of set theory lacks a set that should exist as
much as the
> |alleged uncounted real should exist.
> If by model you mean a set with an epsilon relation on it,
then
> this is correct, but people often mean by model either a
set *or*
> a proper class with an epsilon relation on it. The
cumulative
> hierarchy does not lack a set that should exist-- it
consists
> by de\finition in all the well-founded pure sets.
This is really hard to discuss non-circularly. The words
structure,
class, function, set, collection, relation all have
de\finitions in
the theory and to use the same words outside of the theory is
begging
for confusion. What do you want to take as given? We could
have a
third person in the game, and have the third person start
talking,
saying a in M, aea in A, ain M, \
aÕeain A, aeain E, \
aÕea
in A,
aÕin M, \
aÕÕeaÕin A, \
aÕÕeain A, \
aÕÕea in A, \
aÕeaÕin A,
aeaÕin
A, aÕÕin M, \
aÕÕÕeaÕ\[CapitalO\
Tilde]in A, \
aÕÕÕea in A, \
aÕÕÕeaÕ\[CapitalO\
Tilde] in A,
aÕÕÕeaÕ\[Capital\
OTilde] in
A, aeaÕÕin A, \
aÕeaÕÕin A, \
aÕÕeaÕÕ\[CapitalO\
Tilde] in A, ... and but
where the
third person chooses freely whether to say xey in A or xey in
E,
the problem is that the third person might never shuts up for
the game
to start, but the A-team and the E-team could agree to make
assumptions and act as if the third player said something. The
teams could look at the formula F and as long as they are only
concerned with a winning strategy for ALL models, then the
teams can
consider
the fragments of the formula that always have a winning
strategy in
any \fixed model (a game played after the third player shuts
up) and
ask themselves if A-team wins this fragment O in the yet to be
determined model we eventually get, will one of us be then be
able to
always win the whole formula F?, and if so mark the formula
O-A =>
F-A if A would win and O-A => F-E if E would win. Then if
they ask
themselves if E-team wins this fragment O in the yet to be
determined
model we eventually get, will one of us be then be able to
always win
the whole formula F?, and if so mark the formula O-E => F-A
if A
would win and O-E => F-E if E would win. Assuming it was
possible to
mark the formula each time, then you consider the four cases:
If F is marked O-A => F-A and O-E => F-A then F is false.
If F is marked 0-A => F-E and O-E => F-E then F is true.
If otherwise marked, then F if neither true nor false because
who wins
clearly depends on what the third person says.
If the teams canÕt agree that winning one part implies
another, the
itÕs not really considered bad, because this is the original
case with
proof-theory, because the F that are true or false correspond
to the
sentances N or T where T is either a provable theorem
(without using
/ or //) or the negation of a such a theorem and N is the
IF-FOL
translation of the negations of the second order axioms of
which T or
~T is a provable theorem. The only purpose therefore of
de\fining a
model more precisely as a \finished whole is to conisder the
truth or
falsity of an UNprovable statement. So if makes sense to have
a
type based theory where \first you consider all the provable
statements, and then you could consider things built out of
them (let
the provable statements BE the universe of discourse), I
myself would
be interested in considering a model where the elements of
the model
itself was all the provable statements of the form T=Ex Aa
aex <=>
S[a] where N or T is true as de\fined above for the N
correponding
to the set theory axioms, and then saying T1=Ex Aa aex <=>
S1[a] and
T2=Ey Ab bey <=> S2[b] means we assume that (T1 e T2) in E if
Ex (Aa (aex <=> S1[a])) & S2[x] is a provable true theorem as
de\fined above, and (T1 e T2) in A otherwise. That would be a
most
intriging model to me. But I havenÕt checked to see if \
itÕs a
model
where the usual theorems are true, but itÕs a boot-strapping
kind of
model where you know what everything actually is, I hope it
wasnÕt too
abstract for you.
As far as IÕm considered a model can be anything whatsoever
as long as
the players agree what a, b, etc. are in the model and
whether aeb in
A or aeb in E is to hold for each a and b in the model.
I donÕt understand your claim about the cumulative \
hierarchy,
once you
\finish the model, someone can take the standard interpretation
and
say that some sets are missing, isnÕt V=L considered
restrictive by
mathematicians?
> [...]
> |And once you extend the de\finition of set to have
non-excluded
> |middles, the standard proof about the lack of a set with a
speci\fic
> |property goes away, the theorem becomes the graph of the
bijection
> |between a set and itÕs power set does not have an \
excluded
middle,
> |even if it exists. And yes we can make a hierarchy based on
> |equivalnce classes of sets based on graphs with excluded
middles,
> |itÕll be just like cardinality theory if we do it right.
> The proof that a set does not have a bijection with its own
power
> set works \fine in intuitionist logic, which \
doesnÕt
incorporate
> the law of excluded middle as a rule of inference. If
f:X->P(X)
> is a function, then {x in X : x is not in f(x)} is a set
that is
> not in the image of f. If f(y) = {x in X : x is not in
f(x)}, then
> y is in f(y) if and only if y is not in f(y). That is
inconsistent
> in intuitionist logic.
IÕve never studied intuitionist logic, and I \
donÕt even know
what
inconsistent is de\fined to be in intuitionist logic. Assuming
y is
in f(y) is true leads to ~f(y) is in f(y), which is a
contradiction. Assuming ~y is in f(y) is true leads to ~~f(y)
is
in f(y), which is a contradiction. So we are lead to conlude
that y
is in f(y) is neither true nor false. This is a problem if
you have
an excluded middle, but otherwise, whatÕs the big deal. I
suspect
from talking to you that intuitionist logic DOES have an
excluded
middle, but that it has a limited power to discuss that fact
and a
limited ability to infer from that fact.
> I think youÕre avoiding this conclusion by considering it
possible
> for some statement to be true if and only if it is not
true. I
> understand how IF-logic permits such a thing to occur, but
itÕs
> not convincing to me that this makes good sense.
And I donÕt understand what IT MEANS to not have an exluded
middle and
disallow that. We are coming from different worlds and I
donÕt know
the basis for your ideas, while you know that my meanings are
derived
from the semantics of games. So itÕs a bit unfair for me to \
be
explaining things to you. And there is going to be huge
problems
because we de\fine implication differently, I use a stronger
version
than use, it is more expressive and says more, but therefore
is has
fewer rules of inference. So my biconditional says A <-> ~A
means
that
(~A or ~A) & (A or A) which is logically equivalent to ~A & A
which
means that A cannot be true or false, full stop.
> Once we have a logic with three truth-values, true, false
and
> indeterminate, I donÕt see how it can be invalid for me to
start
> talking about which of the three bins a sentence falls
into. The
> claim that a certain sentence simply fails to be true,
i.e., is
> either false or indeterminate, appears to make logical
sense. ItÕs
> just not a claim that can be expressed in IF-logic.
Huh? Truth is about a winning stratgey in all models, same
with
falsity. Being neither can mean totally different things. It
could
mean that it has a winning strategy for one team in one model
one for
the other team in another model, or it could mean that there
is a
model where the sentance has no winning strategy for either
side. Why
does it make sense to lump these cases together? What we care
about
is truth, which means winning in all models. For instance if
N is the
negation of an axiom and T is a theorem of the axioms, then N
or T
is a validity, true, true in all models, in all worlds. And
there are
TWO ways to negate that CONCEPT, to say false in all worlds
or to
say not the case that it is true in all worlds. IF-logic does
the
FIRST, because the point is that you write SENTANCES and then
you
assert that their truth means something about THE DOMAIN OF
DISCOURSE,
this is what we do in physics, we state the world is such
that T is
true, itÕs what philosophers do. To discuss anything else
invovles
actually quantizying over possible worlds, which I didnÕt
think people
were still serious about doing.
> This is really what I want Hintikka to tell me: why am I
mistaken
> when I think IÕve made an assertion such as the domain of
discourse
> is \finite that is true *exactly* when a certain sentence of
IF-logic
> fails to be true. In what way is this an incoherent
sentence? It
> seems as though he simply ham-strings his theory to make it
impossible
> to say certain things in it.
ItÕs an funny difference of opinions because you think he
ham-stringed
his theory, and it seems to me that you want him to
ham-string his
theory when he hasnÕt. Validities are what we care about,
things true
in all worlds, when you look at a sentance S and say I want to
describe a universe where this is true, then if the sentance
is oFOL
then you can write the validity ~S or S down consider the
\first part
to be a statement of the world of discourse and a latter part
a
theorem of the universe. If thatÕs all you want to do then
you donÕt
need IF-logic, you could pick some \first order axioms and
write N=~A1
or ~A2 or ... or ~An and then be assured that for every
provable
theorem T, the statement N or T is a validity, and so you
could if
you thought the axiom system A1 & ... & An described the
universes of
discourse you had in mind consider that to be a description.
The
point of IF-FOL is to have a stronger version of N where is
is not the
case that N has have a winning strategy for the A-team in ANY
model,
but where the validities N or T can still describe all the
universes
where every T has a winning strategy for the E-team for every
validity
N or T and in THAT sense N is a description of the worlds
under
consideration. ItÕs a much much stronger statement about
conditionality, the whole point of IF-logic is to have
stronger more
expressive statements.
(~A or B) is stronger than A=>B
~A is stronger than it is not the case that A is true, so we
can
say things that you canÕt otherwise say in FOL.
ItÕs is a WEAKER claim about the universe of discourse, it
merely says
consider the worlds where the theorems are true not consider
the
worlds where the (second order) axioms are true. You want
nonsesnse?
It IS nonsense to say consider a FO universe of discourse
where the
SO axioms are true if you just want the FO axioms and oFOL
theorems,
then IF-FOL is useless. If you want to translate SO axioms
into FO
language without assuming a universe of discourse that
INCLUDES SO
entities. Then instead of considering the universe where all
the
axioms are true you should instead consider the universe of
discourse
where all the theorems are true, do you really not get it?
The IF
logic claim is more powerful because when you actually
translate SO
axioms into IF-FOL you get statements that do NOT have
contradictory
negations in second-order logic, that is because they simply
do NOT
have winning strategies for either side, so INSTEAD of
considering
universes where you can VERIFY the truth of the axioms, you
consider
the ones where it is impossible to VERIFY the negation of the
axioms.
It is simply astonding that the ordinary proofs of theorems
carry over
to IF-FOL validities because it is a much much stronger claim
to say
that the theorems are all true in models where the axioms are
not
false, then to MERELY say they are true when the axioms are
true,
because the INTENDED axioms are NOT true in any model. IÕm
trying to
explain why what you are asking for is nonsense, but I donÕt
know if
you can see it.
> Wittgenstein gave an example of an incoherent description:
when itÕs
> \five oÕclock on the Sun. He imagines someone \
asking, what
does that
> mean? and the answer is, Just like what it means to be \five
oÕclock
> here-- but on the Sun. I can imagine that some of the
things I think
> and talk about are confused in sort of this way: it only
seems to me
> that IÕm considering a well-de\fined \
proposition. The only
way I can
> see how it would make sense to consider a system like
IF-logic to be
> ultimate is if I were confused in such a manner about the
things that
> I say that appear not to be expressible in IF-logic.
The claims you want to make (contradictory negations of IF-FOL
formulas) are actually either IF-FOL formulas (in the special
case
where the original formula was actually logically equivalent
to an
oFOL formula) or the negations are ACTUALLY SO statements and
they
require a CHANGE of the universe of discourse to INCLUDE SO
entities.
If you want to stay FO, then IÕd be hard pressed to come up
with some
stronger and more expressive logic than IF-FOL, but maybe
there is
one. I think Hintikka had an extended IF-logic.
> [...]
> |I can grant that no deception was intended if you think it
was just a
> |technical inaccuracy, IÕm a bit scarred (as in maimed, \
not
as in
> |afraid) in that I still do not know the order in which to
resolve
> |things, but IÕm still hoping this book of \
QuineÕs IÕm
reading will get
> |everything in the right order.
> It seems a bit unlikely that there was an intention to
deceive, although
> those of us reading you on sci.math have no \first-hand
knowledge of it.
> I think it will be mighty dif\ficult for you to order your
development
> unless you admit the necessity of starting informally, for
at least a
> brief time, or you decide to go formal all the way and not
care about
> such things as whether the theory has a model.
Which theory are you talking about? Quine is discussing
statements,
and Hintikka has games, either of those seems \fine to treat
informally, they are both just abstractions of VERBS, things
I can do
personally. And so it is disprovable and subject to
experimental
observation ultimately.
> Lorenzen likened the process to the process of building a
ship at sea.
> With one ship, you can build another one inside of it. But
if you are
> just tossed out into the ocean, with some building
materials, you have
> no choice but to learn to swim, and then maybe build your
system while
> swimming.
> Figure out what original unde\fined terms you are willing to
accept.
> Then decide what axioms about them you are willing to
accept. Then
> go from there.
IÕm trying to do that, but I no one is de\fining \
set theory, I
could
learn to swim and then I could build something, but how do I
know
what I personally build is a boat?
> [...]
> |I think this is
> |about describing separation badly, I donÕt think \
itÕs
about logic or
> |language at all, so is EF ~ (Ax Ey Az (zey) <-> (zex and
0=F[z])) a
> |good descrpition of what is intended by separation or not?
I still
> |donÕt know.
> What is intended by separation being false, do you mean? If
IÕm not
> misunderstanding your notation, this is what I remember as
being the
> second-order separation axiom (negated).
Yes, negated, thatÕs the form we want because we or the
negations of
the axioms together to get N so that for a closed oFOL
formula T we
can assert N or T is true to assert that something is a
theorem.
What are the other second order axioms? Do we need a second
order
foundation? A second order replacement (substitution,
collection)?
> [...]
> |> DonÕt confuse rigor with formality. Only a formalist
needs to have
> |> formula de\fined *inside* of a formal theory separately
from
outside.
> |IÕm unfamiliar with your de\finitions of \
realist, formalist
and
so
> |on, it just wasnÕt covered in my education. Some people \
on
this
> |Usenet group have told me to take a set theory class, I
have, they
> |didnÕt cover those terms. Are they covered inmost classes
and was I
> |just unlucky?
> I donÕt know of many places offering courses that would
cover it.
> Philosophy of mathematics is not a big \field, and I \
wouldnÕt
be at
> all surprised if you went somewhere where there simply
werenÕt any
> courses in it. You could try something like the Benacerraf
and Putnam
> book.
Is that book consistent with the de\finitions you were using?
> |IÕm \fine with IF-logic saying that some \
string represent
well-de\fined
> |games and that some games have winning strategies for one
side, and
> |some for the other, and some donÕt. IÕm \
\find with someone
making a
> |claim that the set theoretical universe is such as to make
the string
> |(~A1)or(~A2)or...(~An)or(T) true (when ßeshed out with the
right
> |axioms for A1 through An and the theorem for T,
> Actually, the chunk V_a of the set-theoretical universe I
was referring
> is supposed to make these negated axioms ~Am fail to be
true. Of course
> one canÕt say simply fail to be true in IF-logic. The \
string
> (~A1)or(~A2)or...(~An)or(T) is supposed to hold true in
_all_ domains
> of discourse, since for all of the ones other than the
intended chunk
> of the set-theoretical universe, one of the (~Am) holds.
And for that
> intended one, (T) does.
Close, (~A1)or(~A2)or...(~An)or(T) is a validity (true in all
models)
if T is true in all models where (~A1)or(~A2)or...(~An) is
NOT true,
even if (~A1)or(~A2)or...(~An) is neither true nor false in
that
model. In fact there is no proof that a model exists where
(~A1)or(~A2)or...(~An) is actually false. So IF-FOL
conditional (and
negation) are STRONGER than implication or contradictory
negation.
The statement (~A1)or(~A2)or...(~An)or(T) is a validity is
stronger
than the oFOL statement A=>T for a FO A (and a SO A cannot
even be
stated in oFOL).
> | and if they say that
> |itÕs true for all theorems, then that just leads to the
question what
> |are the theorems, but that isnÕt confusing because we are
forever
> |talking about actual games and these questions are about
what elements
> |can be selected for substitution in the games and which
atomic
> |sentances AeB are going to be true, and which are going to
be false.
> |ItÕs forever a discussion about the rules of the game, no
in\finite
> |regress into types.
> It seems, then, that we have found a dictionary between
whatever I
> might have to say mathematically that concerns this large
submodel
> of the cumulative hierarchy (the sets having rank less than
the
> smallest inaccessible cardinal) can be translated into a
language
> that you can understand!
IÕm still not sure that there is such a submodel that \
\fits the
standard interpretation. If I look at the set theory
validities,
i.e. the valid formulas N or T where T is an oFOL closed
formula and
N is the IF-FOL translation of the SO negations of the SO set
theory
axioms SO alternated together, then it can be true in all
models, but
that doesnÕt mean that there is a model where N is actually
false. It
just means that N or T is a validity.
> |This is FINE for physics because in physics we ALSO play
veri\fication
> |and falsi\fication games in the laboratory, so I can make
them match
> But you donÕt play uncomputable strategies against the
universe,
> so far as I know. IF-logic only works the way that it does
because
> one is implicitly assuming the possibility of using
uncomputable
> strategies.
Where did computable come from? The universe itself plays the
part of
the initial falsi\fier, the existance of the winning strategy
means you
can defeat the universe at the game, every time. If you can
proof the
validity, then you can show that you could win the game N or
T in
any model. So IÕm not making the assumption that the proofs
construct
all the validities, so why should I? And you are totally
forgeting
that the game is usually a premise as well as a conclusion,
so to say
that something is continuous, you might be more precise and
say that
the game Ax Ay (~(0<y) or Ed Ar ~(|r|<d) or |f(x+r) - f(x)| <
|y|))
is true for f, and the meaning of the sentance is that there
is a
winning strategy for the game, you could THEN use that
strategy to
prove other things about the function, like that Aa Ab (a<b &
f(a)<f(b)) => (Az ~(a<z<b) or (Ec a<c<b & f(c)=z)), you \
donÕt
have to
CONSIDER what the word computable means to do any of this.
Validities is based partly on the fact that an oFOL statement
T always
HAS a strategy for one side, so then you can assume a
strategy for one
side and try to prove that a strategy exists so that the
E-team wins
the game N or T and then you assume a strategy exists for the
other
side and try to prove that a strategy exists for N or T for
the
E-team. The word computable (whatever it means) never comes
up, the
fact that T is oFOL and has a strategy for either side is
useful.
Maybe you are bothered that there are two subjects, logic,
the study
of formulas, and set theory the study of sentances T such for
a \fixed
N (unkown to me) the formulas N or T are validities. Set
theory
needs logic, it really does, I didnÕt think this was a
problem.
> How else can it be true that either for every x, ~P(x), or
there
> exists an x such that P(x)? The only way to win that game
is to
> be able to search the whole domain of discourse to check
whether
> any of the elements satis\fies P!
I donÕt know what P(x) is, but saying Ax ~P(x) is true MEANS
that if
you searched the universe that ~P(x) is true for every x, Ex
P(x) is
true means that some choice of x makes P(x) true, whether [Ax
~P(x)]
or [Ey P(y)] is a validity or not is just a matter of whether
P(z) is
true or false for every z, if it is, thatÕs enough, then you
know itÕs
a validity. The point of a validity is that since it is true
for any
play of the game in any model, it is subject to disproof,
thatÕs the
best you can hope for. Proof in science is impossible, so
having a
verb that serves as capable of disproof is the best you can
hope for.
You canÕt perform every experiment in every place at every
time, but
since the claims are universal, you can attempt disproof to
your
heartÕs content.
> |The
> |universe is such that when I do this experiment I get this
kind of
> |results, and one can make DIFFERENT models to help CHOOSE
new things
> |to TEST (in both math and physics). And based on the
results, you
> |might decide to change the rules of the game (new axioms),
or just
> |make new de\finitions to make existing questions easier (for
people) to
> |ask, verify, or falsify.
> This story holds equally well regardless of what axioms we
choose
> to use, however. In fact, if all you care about is
generating
> testable predictions, you should refrain from worrying
about such
> things as whether the mathematical axioms have a model of
the kind
> people think they do. Just deduce consequences! Seriously.
You have to make a choice about WHAT to deduce consequences
FROM, I
still donÕt have a basis to start with. Circular \
de\finitions of
separation arenÕt good, and I donÕt understand \
what the
correct theory
is supposed to be, how do I know that what IÕm deducing
consequences
from is the same thing (or equivalent) to what everyone else
is?
> [...]
> |The physics classes IÕve taken I can teach myself, why is
math so into
> |hiding things? I couldnÕt honestly teach set theory \
today,
even a
> |basic one, because I havenÕt seen a logical presentation.
My physics
> |teachers would answer questions when the students got
together and
> |demanded resolution (like when we asked to know how you
know when to
> |treat a stick as single object versus each molecule like a
separate
> |object, versus each atom as an object versus electrons and
nucleuses
> |versus electons and quarks and gluons), and they did so in
> |non-circular ways.
> Almost nothing youÕve asked about has been about the \
logical
> development of the subject. (Do you have any questions of
the
> form, Is X a theorem of Y? or Is X an axiom of Y?) There
IÕve asked many times what the axioms are. What are the
intended
axioms of set theory? Is separation one of them or does some
SO
replacement and regularity take care of it, or is limitation
of size
considered better? I apologize for being unclear on this, \
IÕd
love to
know the correct axioms, the ordinary FO axioms seem to have
bad
models, so better ones are \fine with me, IÕm not \
sure that the
new
ones IÕd get are better but how can I know until I see them
\first?
> has been scarcely any strictly mathematical (as opposed to
> philosophy of math) question in this discussion. You ask
things
> like How do theorists know that the SEQUENCE to generate h
> [PlanckÕs constant] exists in ZF? that \
doesnÕt have any
clear
> meaning.
Now you canÕt hold me responsible for whatever other people
bring to
the discussion. I would wonder if ZF is the right theory in
which to
create a model that PREDICTS the value of h [PlanckÕs
constant], but
as far as IÕm considered the value we measure in the lab is \
a
rational
number. The point is that people assert that everything is in
set
theory, but they go OUTSIDE set theory to assert that, and I
donÕt
know WHERE that brazen con\fidence comes from. ANYONE can just
ASSUME
that all sets are in their theory. HereÕs an axiom:
Axiom of assertion: For all x, then you assert the standard
interpretation that all sets exist, which cleary includes
every
subset of every set. If I make it this extreme then everyone
knows
itÕs silly, everyone admits that there is more than one
axiom, and the
burden of proof is on the believers in the standard
interpretation to
say the every set exists in the intended model, not to say
well we
have every set, see the Axiom of Assertion, so just take the
class of
all sets and the normal epsilon class-relation and thatÕs a
model with
every set, thatÕs is SO clearly based on a wild meaning of
the axiom
of assertion that isnÕt really in the axiom at all.
> I donÕt think physics classes and mathematics classes are
different
> the way you think. Over and over, in physics we were taught
> theories that we knew were not quite correct, but which we
were
> supposed to value as useful approximations to the truth,
helpful
> in seeking better approximations later. Our main job was to
work
> with them, in spite of whatever imprecisions were involved.
In physics if they had a wrong theory, theyÕd state at the
very very
beginning that it was wrong, thatÕs very different than
saying that
theorems are true for a whole semester.
> There were a number of places where we had to fudge.
In\finities
> appear in certain places that one just has to accept as
being
> not quite right. The energy in the electric \field of a
charged
> canÕt be correct; the product of the charge and the \
electric
> \field vector at it is unde\fined, because the \
electric \field
goes
WHAT! Those can all be \fixed by being more careful. NO ONE
cares
electrodynamics and I kept asking why donÕt we compute the
trajectory
itÕs harder and no one cares. But itÕs up to \
the people who
care
about something to make it work, the purpose of most physics
classes
is not to teach you how to \find analytic solutions, \
itÕs to
develope
physical intuition so that you can recognize correct results
and \fix
problems caused by doing numerical approximations sloppily
and to get
approximate answers so that you donÕt have to get the true
answer for
every situation.
> small and so on. The notation A << B is tossed around with
gay
> abandon, meaning usually that weÕre supposed to pretend \
that
> (A/B)^2 might as well be 0. The business about the force
law is,
> incidentally, not so trivial to correct.
every course I took, I was told no one cares about the motion
of
looking for approximate answers anyway. These are academic
anyway,
because most instructors will be honest and say that real
applications
solve the equations with computers with approximations
anyway, the
derivations of analytic solutions are just to get practise
taking
extreme versions of the laws and recognizing how different
physical
parts dominate the physics in different regemes, itÕs about
physical
intuition, the PDEs are the PDEs and you solve them
numerically in
practise, itÕs not like we HIDE MaxwellÕs \
equation from
students.
> What exactly is the Dirac delta function? Well, itÕs not
really
> a function, and telling you what it precisely is, is beyond
the
> scope of this course. :-) This willingness to work with an
ill-de\fined
> concept is not just accepted; actual pride is taken by
physicists
> in their willingness to dispense with precise de\finitions \
and
Distribution theory, if we want to have a pissing contest
about who
has better teachers, then I can concede that many many bad
physics
teachers exist, in fact thatÕs what IÕd like \
to change,
thatÕs my goal
to improve physics teaching. But the physics teachers *I* had
were
honest about what they knew and didnÕt know and they would
only say
they were sure when they were, and they would differentiate
between
what the books we had said and what the instructor meant, when
something is a derivation versus a consistenty check, when
something
was an approximation and to what degree it was a valid
approximation.
I had very good physics teachers, and the fact that it could
have been
much better is just a case that physics teaching has room for
improvement. But there is a cultural difference too, so IÕll
discuss
IF-FOL validities to give an example.
Some validites like N or T have proofs but others could just
appear to be true top someone informally, but since you \
canÕt
manually check over all strategies over all models, someone
might act
as if it were a validity, and the fact that disproof is
possible
leads some safe feelings for physicists, since they ALWAYS
operate in
a world where disproof is possible no matter what. So itÕs \
not
uncomfortable to explore the what if and make hypothesis
about the
consequences of the what if, if the consequences of the what
if
lead to things that you earlier didnÕt think to check in the
lab and
the lab backs it up, thatÕs good. If later the math falls
apart, then
you just look for new math, not matter how often the math
falls apart,
the laboratory observations are still there (thatÕs why I
*dispise*
how some labs actually THROW AWAY data based on a computer
saying that
it didnÕt matter, itÕs really bad if someone \
later disagrees
with your
theory or your math because those experiments are dif\ficult \
and
expensive and youÕd have to do them again, nothing should be
thrown
away).
> just work intuitively with stuff. One E&M instructor told
us that
> a mathematician would say that f(x)=e^x does not have a
Fourier
> transform, but that he would say it does have a Fourier
transform,
> but we just donÕt understand what kind of thing it is! We
solved
> the problem of a plane wave incident on a circular obstacle
by
> integrating a function that behaves like e^{i * (x^2+y^2)}
> as x and y go to in\finity. (Where it oscillates rapidly,
pretend
Um, plane waves donÕt exist in the lab, and I doubt they \
ever
will, so
thatÕs all just to test a limiting case anyway \
isnÕt it, so
whatÕs the
big deal?
> Experimental sciences all depend on the arrow of time in a
way that
> is not very often explained. I mention this little gotcha
partly
> because to me it has a family resemblance the alleged
circularity
> involved in using a formal system to prove results about
itself.
> One is forced, ultimately, to make assumptions about the
initial
> state of the universe, just as in mathematics one is
forced, ultimately,
> to assume something about how it applies to the world.
A theory can be used two ways, to assume something about the
physical
state of the universe and test the predictions of the theory,
or to
assume the theory is correct and use the observations to
deduce things
about the earlier (or inaccessible to observation part of
the) state
of the universe, success comes from BOTH holding together
well, and of
course there is a possibility for more than one model to \fit
all the
data. No one in physics can say SR is a true model of the
universe
with any authority, or any model, there can always be another
model,
and we donÕt that itÕs consistent, or even \
that it matches
experiment,
but weÕre more clear about the assumptions and the
consequences than
my math classes were about set theory.
> Little bits of slop everywhere. Please note that IÕm not
complaining
> about the courses I took or the instructors I had. I think
they
> did a \fine job overall, and they were right to expect us to
put
> up with some of the messy bits of the subject, basically
not to
> get too hung up on the parts that have to be fudged. We were
> expected to keep going in spite of it.
If they were honest about what was legitimate and what was for
reassurance or other touchy-feely reasons, thatÕs \
\fine.
Confusing one
with the other on their part isnÕt smart though.
> The difference in this respect is the in mathematics, we
come ever
> so much closer to getting it exactly correct, and
ironically get
> to suffer as a result. People come to expect a much higher
degree
> of exactitude. If you were not traumatized by the inability
of your
> physics courses to explain *precisely* how we are supposed
to get
> away with all this stuff, but were traumatized by your
mathematics
> doing so-- the only explanation I can \find is that you
approached
> the two with very different expectations.
I read my physics textbooks years before taking my physics
classes,
and with about 3 years more than the math prerequisites so it
was
usually the case that I could tell what was actually intended
from
reading the book, and if a book annoyed me, then IÕd
supplement it
with another book. There are lotÕs of good physics books \
that
start
at simple parts and go far, but most math books seem to not
want to
start at the beginning, but instead with set theory, which
makes the
latter subjects easier if you assume set theory, but there
just
werenÕt good set theory books that I could \find. \
You also have
to
remember that reading physics textbooks years early and
taking math
classes with the Moore method are about as different as night
and day.
> Now, the one thing youÕve suggested is that at least one \
of
your
> mathematics professors failed to acknowledge the
imprecision in
> his remark about the axiom of comprehension. I would have
to hear
> just exactly how that exchange went between the two of you
before
> I would assume that this is so. If he did pretend to be
absolutely
> precise when he was not, however, IÕm sorry that he did.
I could simply erase the incident from my mind if I just knew
what to
replace it with, I still donÕt. I want some authority that a
particular theory IS set theory, making up my own axioms
doesnÕt
tell me that what IÕm doing is the set theory everyone else \
is
doing.
> I wouldnÕt say that itÕs much worse than the \
physics
professor who
> once was diagonalizing a matrix in class, and did it wrong.
I pointed
> out his mistake, and he muttered that I couldnÕt have
\figured it out
> like that, erased his work, and then did it over more
carefully.
I was called (by my friends) into physics classes to sit on
on their
classes to call their professors on their mistakes, the only
time I
failed was when a professor started where they left off
yestersay
(before I sat in) in a class I hadnÕt taken, so it took \
about
45
minutes for me to recognize that that isnÕt a wave. People \
are
sometimes sloppy, you shouldnÕt just trust instructors but
should you
should attempt to understand everything they say (if the
subject is
important to you) and all the reasoning and justi\fication, \
just
trusting someone to be right is bad news. My frustration
about math
is about AXIOMS. If I canÕt \find uncircular \
books to back up my
teachers (and wasnÕt allowed when taking the course) and the
books
arenÕt clear and the teachers werenÕt clear \
about the
de\finitions then
I donÕt even know what the subject IS, so to compare that to
a physics
class being sloppy about a momentum-eigenstate in a (rigged)
Hilbert
space or a plane wave solution or something isnÕt fair at
all, itÕs
more like if your physics class said that SR was based on the
assumption that the speed of light is Ôtrueis \
all *mumble*
frames.
ThatÕs preposterous, and just because you can speak
informally to
disguise sloppiness doesnÕt make it better, it should be
\fixable. I
should be able to \find out what the axioms are, full stop,
this isnÕt
supposed to be a power-trip or a head-game. In physics I can
\find
books and experiments to make sloppiness clear, usually
bringing some
more math to the table \fixes things in physics, and more math
might
\fix the problems in math too, but I canÕt \
\find books to \fix it. I
could make up my own axioms, but that wouldnÕt be set \
theory,
itÕd
be something theory. To know what set theory is is like
knowing
what SR is, I have to have the de\finitions or the consequences
one
or the other.
> |But the math proffessors just say why are you so
> |interested in foundations, I thought you liked physics?,
IÕm just
> |simply tired of being discriminated with, IÕm certain \
that
they talk
> |not in circles amongst themselves,
> IÕve never had a lot of conversations with set theorists,
but I can
> assure you, conversations between professors of mathematics
are no
> more formal, rigorous, and ultra-precise than their
conversations
> with students. They are much less liable to be tripped up by
> inadvertent imprecisions of the kind youÕve been
complaining about,
> and consequently they take less care to avoid them amongst
themselves.
> The kind of stuff youÕve been agonizing about is just not
worried
> about! When writing papers, of course, thereÕs a third
standard that
> is in some ways more formal, but also allows for details
that the
> (professional) reader doesnÕt need to be left out.
There should be some source SOMEwhere that is clear about the
de\finitions of the \field, shouldnÕt \
there?
> |and I think itÕs down right rude to
> |hide the actual logical developement from people just for
not being
> |in the club, this isnÕt middle school this is science.
> Perhaps you think that at some later date, the students who
mean
> to go on to become professional set theorists get taken
aside and
> have the real development of the subject taught to them. I
donÕt
> think so.
Then how do they know, IÕve seen lotÕs of \
books at a naive
informal
level and many advanced level books that assume you already
know
things not covered in the informal books, where are the course
materials for the intermediate level courses, where are the
books of
intermediate level?
> | I consider
> |math to be science.
> Then try treating it like one.
> The key objections to be made to a physical theory are that
its
> predictions are observed to be incorrect, itÕs been
superceded by
> a more comprehensive theory, and that itÕs needlessly
complex.
> The corresponding objections can be made of a mathematical
theory
> too, in principle, but I see you making an awful lot of
objections
> that are of a completely different kind.
In IF-FOL I can imagine an observational refutation of a
proposed
validity (by demostrating game play inconsistent with that
description), but I donÕt see how you can observe other
things to
invalidate a theory, the theorems are treated formally if you
object
to the English, and then someone will continue using the
English, so
for most people itÕs just a waste of time to observe \
anything
in
math.
> Allow it to be just as sloppy and messy as physics is and
nearly
> all of those remaining troubles are gone! If you are still
of the
> impression that you canÕt tell what ZF is, try naming some
statement
> which you either have trouble knowing how to formalize, or
donÕt know
> whether itÕs an axiom, or a proof which you \
donÕt know
whether itÕs
> valid, and we can surely clear it up for you.
Is Ax ~Ef (Ea aex & {0,{0,a}}ef) & (An Ab (bex =>
({n,{n,b}}ef => (Ec
cex & {nU{n},{nU{n},c}}ef & ceb)))) the foundation axiom? Is
it
considered part of set theory? I have more questions like
this, but
for all I know IÕm already kill-\filed by \
everyone.
> [...]
> |There is the translation from English to set theory and
back
> |constantly, there is no way for me to tell that IÕm doing
it the same
> |way. The point is that someone can say X is a theorem: Y,
QED but
> |if you donÕt know WHAT it is a thoerem of, then I \
canÕt
turn around to
> |the guy next to me and say X is a theorem because if IÕm
asked
> |theorem of what then I canÕt answer, so the theoremhood \
of
the
> |statement isnÕt really proven (I canÕt \
carry it around
with me or
> |apply it outside of the set theory class), it was only
stated as a
> |theorem of SOMETHING. Only after we know the axioms can we
know that
> |X is IN FACT a theorem of THAT axiom system.
> We donÕt depend upon axiom systems in the way that you
imagine.
> When someone claims that a result is a theorem, they mean
that it
> has been proven, period. End of sentence. Not, proven in...
but
> proven.
What? You donÕt seriously mean proven without assuming any
standards
of proof or axioms, do you? You could always say (N or T) is a
validity, instead of saying T is a theorem, and thatÕs be \
more
clear, but then IÕd probably be happy because then someone
would STATE
what N is, which is the big thing I donÕt know. If somehow
you handed
be an algorithm to compute all the Ts, then I could postulate
an N,
but if I donÕt have all the Ts and I donÕt \
know N then how
can I make
my own theorems, or recognize a theorem T when I see one?
> It is true that having proven a result, your proof can be
examined
> to see what kinds of axioms would suf\fice for it. If you ask
what
> itÕs a theorem in, you are liable to get the answer ZFC, \
not
> because the proof has been examined in this way, but just
because
> itÕs so rare for any other assumptions to be used. If one
does use
> an assumption not supported by ZFC, itÕs expected that one
will
> mention it somewhere in the course of the proof.
But what IS the ZFC axiom system? Does it have separarion?
Collection? Replacement? FO? SO? Foundation?
> [...]
> |> I had in mind the common situation where one has a
\first-order theory.
> |> In that case, we have the Goedel completeness theorem
that says the
> |> logical consequences of a set of axioms are the same as
the
consequences
> |> that can be deduced using standard \first-order logic.
> |Are you sure about how you stated that? IÕm assuming
standard
> |\first-order logic is ordinary \first order logic \
(so not
IF-logic or
> |SOL), but with IF-logic you can make \first order statements
that
> |arenÕt statements of ordinary \first order \
logic, so you
claim seems,
> |... a bit sensational. If itÕs true, then \
thatÕs great,
but I want to
> |know if thatÕs what you meant.
> Hintikka misleads the unwary reader by causing him (i.e.
you) to
> think that \first order statement has come to mean something
more
> than a statement in ordinary \first-order logic. It has done
so
> only in HintikkaÕs own terminology. He claims his system
should
> count as \first-order logic, but this is not what anybody
else means
> by it.
I have no idea what anyone else means, Hintikka has formulas,
they
have quanti\fication of individuals of the same type
(individuals of
the domain of discourse) thatÕs FO. SO involves formulas \
that
quantify over a second type, some SO formulas are logically
equivlanet
to oFOL formulas, others to IF-FOL formulas, others to
neither. But
IF-FOL doesnÕt assume the existance of another type.
> Certainly the Goedel completeness theorem is only for
\first-order
> logic, not IF logic.
> Keith Ramsay
I would still be unclear what you mean, youÕd have to \
de\fine
logical
consequence and deduce with oFOL, and IÕm unclear about the
point
too. IF-FOL has stronger expressions than implication. FOL is
weak
because you have to know that the axioms are TRUE in a model
before
knowing that the theoresm MUST be true, but with IF-FOL you
only need
that the axioms not be false to assert that the theorem MUST
be true.
===
Subject: Re: SkolemÕs Paradox and why is math the way it is?
> You missed my point, IÕm saying that if we eplain axioms
badly, then
> people could interpret the axioms in two ways (one
contradictory, the
> other not) and that if other things are de\fined in terms of
the axioms
> then the de\finitions would be different, and then two people
could
> have the case that one comes to a contradiction and the
other doesnÕt
> and they could trace this DISagreement BACK to a
disagreement about
> the interpretation of the axiom.
It is unambiguously de\fined which sentences are and which are
not valid
consequences of FOL+ZF(C) (and this is not so for IF+ZF(C),
btw).
So there cannot be such differences of opinion. What can
happen is that
we can not prove nor disprove some sentence. That may be
annoying, but
canÕt be helped.
> With IF-logic, there is an informal level, itÕs about
winning
> strategies, I can handle that because I know what that
means, and that
> assigns a meaning to every formula as being about a game
played in a
> model (which is IMO just an agreement with both sides on
what the
> universe of discourse is and about who wins a game of xey
for evey x
> and y in the universe of discourse). Nothing more, and
nothing less.
> When I was reading about the foundation axiom I saw a proof
that there
> is no set of all games, I thought that was an interesting
fact.
But FOL is a subset of IF-logic, so all FOL sentences also
have this
informal, intuitive meaning, using game- or dialog-like
semantics.
Btw, using games as an alternative for sets has its own
problems. Do you
know the Metagame paradox, for example?
--
Herman Jurjus
===
Subject: Re: SkolemÕs Paradox and why is math the way it is?
> You missed my point, IÕm saying that if we eplain axioms
badly, then
> people could interpret the axioms in two ways (one
contradictory, the
> other not) and that if other things are de\fined in terms of
the axioms
> then the de\finitions would be different, and then two people
could
> have the case that one comes to a contradiction and the
other doesnÕt
> and they could trace this DISagreement BACK to a
disagreement about
> the interpretation of the axiom.
> It is unambiguously de\fined which sentences are and which
are not valid
> consequences of FOL+ZF(C) (and this is not so for IF+ZF(C),
btw).
> So there cannot be such differences of opinion. What can
happen is that
> we can not prove nor disprove some sentence. That may be
annoying, but
> canÕt be helped.
You missed my point. You have to \first unambigiously STATE the
axioms
of FOL and ZF(C), I havenÕt gotten straight answers about
what the
axioms of ZF(C) are, so until that point, the consequences of
hidden
things are hidden. So is GoedelÕs result a valid consequence
of
FOL+ZF(C), or how about any result that dependsing on
Con(FOL+ZF(C))?
IÕd like a de\finition of a consequence almost as \
much as IÕd
like to
know which axioms are consider the axioms about sets with the
standard interpretation as opposed to models of a FO theory
with a
binary relation e
> With IF-logic, there is an informal level, itÕs about
winning
> strategies, I can handle that because I know what that
means, and that
> assigns a meaning to every formula as being about a game
played in
a
> model (which is IMO just an agreement with both sides on
what the
> universe of discourse is and about who wins a game of xey
for evey x
> and y in the universe of discourse). Nothing more, and
nothing less.
> When I was reading about the foundation axiom I saw a proof
that there
> is no set of all games, I thought that was an interesting
fact.
> But FOL is a subset of IF-logic, so all FOL sentences also
have this
> informal, intuitive meaning, using game- or dialog-like
semantics.
But IÕm told that IÕm a bad person if I use \
any informal
description
other than the informal level of \first assuming all sets
(whatever
that means) and then assuming that the axioms accurately
describe
truths about all sets.
> Btw, using games as an alternative for sets has its own
problems. Do you
> know the Metagame paradox, for example?
IÕve not heard it before (or even stated), the quotes make \
me
think it
might be similar to the twin paradox in that there is
actually no
contradiction, just a more complicated picture than it \first
appears,
are you sure it isnÕt really a paradox about the alleged set
of all
games? I donÕt know since I havenÕt heard it.
===
Subject: Re: SkolemÕs Paradox and why is math the way it is?
Originator: joshp@xoxy.net (joshp)
> You have to \first unambigiously STATE the axioms of FOL and
ZF(C)
Axioms for FOL and ZFC are unambiguously stated at:
<http://us.metamath.org/mpegif/mmset.html#axioms>
--
Josh Purinton
===
Subject: Re: SkolemÕs Paradox and why is math the way it is?
> You have to \first unambigiously STATE the axioms of FOL and
ZF(C)
> Axioms for FOL and ZFC are unambiguously stated at:
> <http://us.metamath.org/mpegif/mmset.html#axioms>
(or just using?) meta-logic, is this another name for SO
logic? Does
that mean that that is a SO ZFC axiom system, or are SO
theories
considered not ZFC by de\finition and if so is there a standard
SO
axiom that goes by a different name? I read a bit of the web
page,
but since you started out introducing it as FO I just kept
getting
more and more confused with the meta-discussions.
===
Subject: Re: SkolemÕs Paradox and why is math the way it is?
> You have to \first unambigiously STATE the axioms
> of FOL and ZF(C), I havenÕt gotten straight answers about
what the
> axioms of ZF(C) are, so until that point, the consequences
of hidden
> things are hidden.
YouÕre extremely lazy, then?
===
Subject: Re: SkolemÕs Paradox and why is math the way it is?
> You have to \first unambigiously STATE the axioms
> of FOL and ZF(C), I havenÕt gotten straight answers about
what the
> axioms of ZF(C) are, so until that point, the consequences
of hidden
> things are hidden.
> YouÕre extremely lazy, then?
There are many issues that people seem to disagree about when
I try to
get a consensus answer to this (I could easily make my own
axioms, but
that wouldnÕt mean they were equivalent to ZF(C), the axioms
are the
only issue where a group opinion matters, but it does). For
instance
are the ZF(C) axioms FO or SO? Which is considered the real
ZF(C)
axioms? Which version?
===
Subject: Re: SkolemÕs Paradox and why is math the way it is?
> There are many issues that people seem to disagree about
when I try to
> get a consensus answer to this (I could easily make my own
axioms, but
> that wouldnÕt mean they were equivalent to ZF(C), the
axioms are the
> only issue where a group opinion matters, but it does). For
instance
> are the ZF(C) axioms FO or SO? Which is considered the real
ZF(C)
> axioms? Which version?
These are completely trivial questions. By looking in the
literature
you can easily become an expert on any minor variants there
are, and
your own judge of their signi\ficance or \
insigni\ficance.
===
Subject: Re: SkolemÕs Paradox and why is math the way it is?
> You have to \first unambigiously STATE the axioms
> of FOL and ZF(C), I havenÕt gotten straight answers about
what the
> axioms of ZF(C) are, so until that point, the consequences
of hidden
> things are hidden.
> YouÕre extremely lazy, then?
IÕve been told different things, some of them untrue, some \
of
them
vacuous (or circular), and others just that contradict each
other.
IÕm now very untrusting, but still not lazy.
===
Subject: Re: SkolemÕs Paradox and why is math the way it is?
> [...]
> |I think the best way to teach quantum mechanics is to
assume that the
> |wave-function is real (exists), and that the equations
describe how it
> |moves, and thatÕs it, in practise thatÕs \
all you need and
every
> |interpretation takes that seriously to the extent that the
> |interpretation takes anything seriously at all.
> I mostly agree, but at least one of my physics professors
in college
> considered treating the wave-function as real as wrong,
wrong, wrong.
> It applies only to a statistical ensemble, not to just a
single
> system. (!) I would feel at least some qualms about leaving
students
> with the idea that thereÕs a consensus opinion among
physicists about
> the reality of it.
LetÕs remind that there is no universally accepted
philosophical
interpretation of quantum physics. If you take Copenhagen
interpretation as true, or any interpretation that attributes
existence to the wave function then you arrive at PenroseÕs
absurd
quantum dualism. Reviving Cartesian Dualism, what a great
idea! Hah!
> Hintikka misleads the unwary reader by causing him (i.e.
you) to
> think that \first order statement has come to mean something
more
> than a statement in ordinary \first-order logic. It has done
so
> only in HintikkaÕs own terminology. He claims his system
should
> count as \first-order logic, but this is not what anybody
else means
> by it.
> Certainly the Goedel completeness theorem is only for
\first-order
> logic, not IF logic.
HintikkaÕs authority on Godel is at best laughable. His book
On
Godel is ridden with mathematical errors, and sloppy
philosophical
arguments. It was the worst book on Godel I read. I donÕt
think he is
an intelligent person. I still got to take it from the
library, and
write here on some of the errors. He couldnÕt even tell what
Godel
numbering is rigorously, yet alone tell the relation between
incompleteness theorems and Turing undecidability properly.
Philosophy departments are not what they used to be.
--
Eray Ozkural
===
Subject: Re: SkolemÕs Paradox and why is math the way it is?
> [...]
> |I think the best way to teach quantum mechanics is to
assume that the
> |wave-function is real (exists), and that the equations
describe how it
> |moves, and thatÕs it, in practise thatÕs \
all you need and
every
> |interpretation takes that seriously to the extent that the
> |interpretation takes anything seriously at all.
>
> I mostly agree, but at least one of my physics professors
in college
> considered treating the wave-function as real as wrong,
wrong, wrong.
> It applies only to a statistical ensemble, not to just a
single
> system. (!) I would feel at least some qualms about leaving
students
> with the idea that thereÕs a consensus opinion among
physicists about
> the reality of it.
> LetÕs remind that there is no universally accepted
philosophical
> interpretation of quantum physics. If you take Copenhagen
> interpretation as true, or any interpretation that
attributes
> existence to the wave function then you arrive at \
PenroseÕs
absurd
> quantum dualism. Reviving Cartesian Dualism, what a great
idea! Hah!
Quite a grandiose claim. I agree that Cophenhagen +
wave-function-reality leads to dualism, and so does any other
interpretation that takes the Copenhagen macro-world
seriously and the
wave-function seriously, but that is hardly the class of all
interpretations that take attribute existence to the wave
function,
far from it. If you attribute existence to the wave function
and
nothing else, there is no dualism of any kind, let alone
silly kinds.
> Hintikka misleads the unwary reader by causing him (i.e.
you) to
> think that \first order statement has come to mean something
more
> than a statement in ordinary \first-order logic. It has done
so
> only in HintikkaÕs own terminology. He claims his system
should
> count as \first-order logic, but this is not what anybody
else means
> by it.
>
> Certainly the Goedel completeness theorem is only for
\first-order
> logic, not IF logic.
> HintikkaÕs authority on Godel is at best laughable. His
book On
> Godel is ridden with mathematical errors, and sloppy
philosophical
> arguments. It was the worst book on Godel I read. I donÕt
think he is
> an intelligent person. I still got to take it from the
library, and
> write here on some of the errors. He couldnÕt even tell
what Godel
> numbering is rigorously, yet alone tell the relation between
> incompleteness theorems and Turing undecidability properly.
> Philosophy departments are not what they used to be.
I like Hintikka because while his works are always riddled
with errors
I can \figure out what he meant and correct it myself a good
percentage of the time, and I *agree* that itÕs SAD that \
that
makes
him BETTER than most other people I read who either talk in
obvious
circles or never go anywhere or assume that I already know
facts from
other books (unreferenced) that I canÕt \find \
anywhere. I think
Hintikka is intelligent, I donÕt know if he has bad editors
for his
books, or if his english is so bad that thatÕs the best his
editors
can do, but even that doesnÕt excuse all his errors because
sometimes
they are in the formulas and so there is no excuse at all,
either heÕs
sloppy himself or some copy editors are really really bad,
and my bet
would be on Hintikka being sloppy, but that doesnÕt make him
unintelligent.
===
Subject: Re: SkolemÕs Paradox and why is math the way it is?
> I like Hintikka because while his works are always riddled
with errors
> I can \figure out what he meant and correct it myself a good
> percentage of the time, and I *agree* that itÕs SAD that
that makes
> him BETTER than most other people I read who either talk in
obvious
> circles or never go anywhere or assume that I already know
facts from
> other books (unreferenced) that I canÕt \find \
anywhere. I
think
> Hintikka is intelligent, I donÕt know if he has bad \
editors
for his
> books, or if his english is so bad that thatÕs the best \
his
editors
> can do, but even that doesnÕt excuse all his errors \
because
sometimes
> they are in the formulas and so there is no excuse at all,
either heÕs
> sloppy himself or some copy editors are really really bad,
and my bet
> would be on Hintikka being sloppy, but that doesnÕt make \
him
> unintelligent.
Let me just say that he isnÕt the right kind of person to
learn Godel
from. IÕve read books of great mathematicians and
philosophers, and I
can tell the difference. Rigor is very important in
philosophy, as
well as in mathematics.
--
Eray Ozkural
===
Subject: Re: SkolemÕs Paradox and why is math the way it is?
> I like Hintikka because while his works are always riddled
with errors
> I can \figure out what he meant and correct it myself a good
> percentage of the time, and I *agree* that itÕs SAD that
that makes
> him BETTER than most other people I read who either talk in
obvious
> circles or never go anywhere or assume that I already know
facts from
> other books (unreferenced) that I canÕt \find \
anywhere. I
think
> Hintikka is intelligent, I donÕt know if he has bad \
editors
for his
> books, or if his english is so bad that thatÕs the best \
his
editors
> can do, but even that doesnÕt excuse all his errors \
because
sometimes
> they are in the formulas and so there is no excuse at all,
either heÕs
> sloppy himself or some copy editors are really really bad,
and my bet
> would be on Hintikka being sloppy, but that doesnÕt make \
him
> unintelligent.
> Let me just say that he isnÕt the right kind of person to
learn Godel
> from. IÕve read books of great mathematicians and
philosophers, and I
> can tell the difference. Rigor is very important in
philosophy, as
> well as in mathematics.
I disagree with your opinion about the importance of rigor vs.
clarity. If you made up your own symbols and your own logic
and your
own axioms and your own rules of deduction, then it doesnÕt
matter how
rigorous it all is if no one understands what you are doing.
And if
someone understands what you are doing and saying, then you
can make a
liberal amount of mistakes (that donÕt undermine your
clarity, for
instnace having text and formulas where the ßow says which is
correct) and the worst that happens is that people think you
were
sloppy. OF COURSE, itÕd be happy-fun-nice if everyone could
be both
clear and correct all of the time, but being clear can make
up for
being wrong (because your readers can correct you while
reading you),
whereas being correct cannot make up for being unclear
because you are
just wasting everyoneÕs time.
Just to be clear, I am NOT critizing other writers on Goedel
of being
unclear, IÕm saying that HintikkaÕs errors are \
obvious, and
IÕd
prefer obvious errors over an unclear exposition with fewer
error or
over a one without errors to the author that is so unclear
that the
*reader* gets errors in their (incorrect) interpretation of
the work.
So IÕm saying that Hintikka COULD be worse, and where I draw
my line,
heÕs on the side where itÕs useful to read his \
books.
I wouldnÕt suggest burning all books not written by \
Hintikka,
in
fact IÕd be against that. IÕd suggest getting \
his point from
his
books, and reading OTHER books to strengthen your background
of the
subject, from the vocabulary Hintikka uses, itÕs absolutely
clear that
that is what he intends the reader to do as well, no one could
successfully read his books without a getting a background in
the
subject either before or after reading HintikkaÕs book.
Maybe we are agreeing, since I donÕt know what you mean by
learn
Goedel from, but your remarks about a kind of person just seem
offensive which makes me what to assume that I have
misunderstand what
===
Subject: Re: SkolemÕs Paradox and why is math the way it is?
> Rigor is very important in philosophy, as well as in
mathematics.
rigour. So not only are you a nonmathematician, but a
nonphilosopher
to boot.
--
Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html
Lacan, Jacques, 79, 91-92; mistakes his penis for a square
root, 88-9
Francis Wheen, _How Mumbo-Jumbo Conquered the World_
===
Subject: Re: SkolemÕs Paradox and why is math the way it is?
> HintikkaÕs authority on Godel is at best laughable. His
book On
> Godel is ridden with mathematical errors, and sloppy
philosophical
> arguments. It was the worst book on Godel I read. I donÕt
think he is
> an intelligent person.
Back to your old standbys of abuse and ad hominem attacks :-(
--
Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html
Lacan, Jacques, 79, 91-92; mistakes his penis for a square
root, 88-9
Francis Wheen, _How Mumbo-Jumbo Conquered the World_
===
Subject: Re: SkolemÕs Paradox and why is math the way it is?
> HintikkaÕs authority on Godel is at best laughable. His
book On
> Godel is ridden with mathematical errors, and sloppy
philosophical
> arguments. It was the worst book on Godel I read. I donÕt
think he is
> an intelligent person. I still got to take it from the
library, and
> write here on some of the errors.
But you wonÕt, right? You said some time ago that you would
point
out several ßawed arguments on my web pages about the
incompleteness theorem.
===
Subject: Re: SkolemÕs Paradox and why is math the way it is?
|Just write down the negations of the axioms, and write down
the
|statement of a proposed theorem, and or them all together,
then:
|
|validity of (~A1 or ~ A2 or ... or ~An or P) <=> theoremhood
of P
|
|IsnÕt that what we wanted all along? I donÕt \
understand how
this is
|ANY different that the normal purpose of the mechanically
DEDUCTIVE
|branch of mathematics, you want to know what necessarily
follows.
I donÕt know how you got the idea that itÕs \
normal to
consider a
branch of mathematics to be mechanically deductive, or why
you think
determining what can be inferred from the axioms is the main
thing
that we want. As I see it, thatÕs putting the cart before \
the
horse.
The primary purpose of axioms is for studying something else,
not for
studying the axioms themselves (although logicians are of
course
interested in that too).
ThereÕs a famous Russell quote where he says something like
that,
but I donÕt consider it quite truthful. Remember that \
Russell
was a
logicist. Also, he was not interested only in \first-order or
formal
deductions. Quite a lot of mathematics can be reasonably
interpreted
as motivated by wanting to determine the consequences of a
*second-order*
set of axioms, which in general canÕt all be deduced by one
mechanical
system.
|And
|with IF-logic you have the AVANTAGE of having your negations
of axioms
|be stronger than in ordinary FOL, because basically the
axioms can
|be Pi.1.1.
Why not second-order logic, which is stronger still?
Maybe the main reason \first-order logic is considered \
important
(in spite of being weaker in expressive power than IF-logic or
second-order logic) is that thereÕs a complete \
axiomatization
of it. When we want to write down a formal system, typically
we
write \first-order axioms for some structure.
Once one starts talking about logics for which we donÕt have \
a
complete set of deductive rules, then it no longer serves the
same purposes as \first-order logic does. The main reason for
not
using all of second-order logic is gone. I donÕt see the \
other
reasons for preferring to use a logic intermediate between
\first-order and second-order as very strong.
Now you discuss strengthening the schema of separation to
something
much closer to the second-order axiom, whose negation is
expressible
in IF-logic.
[...]
|ItÕs negation is EXPECTED to be true for set theory,
|speci\fically THE set theory of the STANDARD interpretation.
Well, doesnÕt Hintikka technically de\fine the \
negation to be
the
sentence corresponding to the game where the players roles are
reversed? His negation of a sentence like this is much
stronger
than what others would say the negation should be. People
would
say that this formula (expressing in IF-logic the failure of
the
second-order selection axioms) *fails to be true* in the
standard
interpretation (i.e., the cumulative hierarchy of sets).
| If it
|were NOT true, then we wouldnÕt KNOW if the reals had more
real
|numbers in them or if it was just a lack of a bijection for
counting
|them that makes the set appear uncountable from within ZF.
Why not? You see, the happy realists among us believe that
all we
need to do to guarantee that {r : r is a real number}
includes all the
real numbers (regardless of where they come from) is just to
SAY it.
Why is such-and-such real number in it? BY DEFINITION.
If I had made an attempt at cornering the world apple market,
and
had a huge warehouse containing what I claimed were all of
the apples
in the world, it would make sense to say, Wait, couldnÕt you
have
missed some of the ones that are harder to get to? But if I
*refer
to* all the apples that exist, to say to me, wait, couldnÕt
there
be apples that you didnÕt refer to? is incoherent. This is a
huge
difference. A reference is completely unlike a container in
this
respect. For a reference to include something, it doesnÕt
need to
physically go out and fetch the referenced object. It just
needs to
be de\fined so as to include it.
|The thing is, IÕm not sure if we can prove that ANY model \
of
the ZF
|axioms exist if we replace separation with the stronger
version above.
Obviously we canÕt prove it in ZF, but we can prove it in
ZF+there
exists a strong inaccessible cardinal. If there is a strong
inaccessible
cardinal x, then the sets having rank <x are a model of ZF
where the
second-order separation axiom holds.
| But we donÕt NEED to do that, if all we are interested in
is proving
|implications, since (A1&A2&A3...&An)->T can be rewritten to
(~A1 or
|~A2 or ... or ~An or T), and then you can have stronger
axioms like
|the stronger than separation axiom I gave above, and you can
try to
|prove that itÕs a validity.
But perhaps one wants more than just to prove such
implications
(at least, more than just implications in \first-order or IF
logic)?
Do you think you have a reason to believe we donÕt care \
about
proving implications in second-order logic?
[...]
|Historically
|mathematicians wanted to retreat to axiomatic set theory to
avoid
|contradictions.
There are lots of sources that present a very sketchy
description
of the history that makes it sound like axiomatic set theory
was
developed largely to block paradoxes like RussellÕs paradox,
but
I think this is misleading.
A very slightly less sketchy description of the history will
note
that Cantor presented an informal set theory without any
now-known
internal inconsistencies in it. He clearly means to
distinguish
between sets and what we would now call proper classes, and
warned
that treating proper classes like sets was a problem. His
original
motivation for developing set theory came from his study of
Fourier
analysis. He developed a theory of cardinals and ordinals
that is
essentially the same as what we use now.
Frege is famous for his book on founding arithmetic on logic,
where his \fifth axiom is self-contradictory, because of the
Russell paradox. But Frege didnÕt make a distinction between
extensions that we would call sets and ones that we would call
proper classes. He turned later in life to the idea that
mathematics could be founded instead on geometry.
Cantor had thought that it was intuitive (!) that every set
could be well-ordered, i.e. that there exists a total ordering
< on it such that every nonempty subset has a least element
under <. Eventually a proof of this claim was given, although
some critics had misgivings on it.
Zermelo then stated his list of axioms in order to identify
what
needs to be accepted in order to accept the proof of the
well-ordering theorem. Among the principles needed of course
is
the axiom of choice, which people (including some critics) had
used without being aware that they were assuming anything.
| So the implication of T1 and T2 (from {A1, ...,
|An}) shows that T isnÕt set-theoretically-inconsistent with
T2. Which
|is all that we knew anyway if we took axiomatic set theory
|seriously. The point (for physics at least) is to use a
collection
|of theorems that are not mutually inconsistent. If thatÕs
too hard,
|then a collection of theorems that are no mutually
|set-theoretically-inconsistent might be good enough, and
IF-logic
|allows that, EVEN IF the axioms ARE inconsistent (lack any
faithful
|models).
IÕm not sure why we need (for the purposes of physics) to be
so
wary. A theory of physics states various premises from which
one can
infer observable results, and then one goes and observes
them. (Or
one observes that they are incorrect.)
There is as far as I can see no need to attempt to *show*
that the
premises are consistent. Obviously, if the theory is using
standard
logic, an inconsistency means that the theory has failed,
because
once one can derive a contradiction from the axioms, one can
derive
anything at all, including predictions that one knows
disagree with
experiment. So certainly one wants the premises to be
consistent.
But thereÕs never any guarantee of success for a theory, and
\finding
an inconsistency is no worse in this respect than any other
way in
which the theory might fail.
ItÕs prejudicial to a theory to say that merely predicting
results
in an economical way is not enough.
[...]
|I know that this could make me unliked (as if I could but
much MORE
|unliked than I already am) because most people will skim
that and
|decided that what I *really* said was [...]
Excuse me for \finding it rather weird how often you suggest
that
people reading this seriously dislike you. You started this
from
just about the \first time you posted. This strikes me as being
a kind of insult to our collective emotional intelligence of
your audience.
There are a lot of people who claim on sci.* newsgroups to be
disliked, and most of them pretty soon start using that claim
as
an explanation for why people disagree with them. I sure hope
that youÕre not trying to set yourself up as this kind of
martyr.
This is actually one of the most irritating things people do
here.
[...]
|But if ZF *is* inconsistent, then any STRONGER system is
going to fail
|too, and since ZF has historical primacy (which should NOT
matter in
|MATH, but because of Platonists and an almost religious zeal
about set
|theory, it DOES appear to matter), then any other system
will either
|be too weak (for not containing set theory properly) or
|inconsistent (for being equiconsistent with set theory),
itÕs just
|an unfair historically-based game in my opinion.
Oh baloney. What youÕre doing in this paragraph is
pre-complaining
about how your as-yet-undeveloped theory is liable to be
received.
This is not a fair criticism of the community.
|HereÕs a way to look at it, that IÕm fairly \
sure will be
misunderstood
|as saying something other than what I intended, but IÕll \
try
anyway.
|Set theory is supposed to have every existentially possible
thing
|such that something is either de\finately in it, or \
de\finately
not in
|it.
Not exactly, no. I donÕt know that anybody claims that the
cumulative
hierarchy includes all such sets. There might be perfectly
well-de\fined
sets that happen not to be well-founded, and there certainly
are ones
that are impure.
| But if a model existed, then you could ask of anything (so
any
|set) is it in the model?
Talking about models seems to me a red herring. There is one
key
model (in the sense of being a proper class satisfying the
axioms
of ZFC), namely, the cumulative hierarchy. If you have qualms
about the class of well-founded pure sets (i.e., the
cumulative
hierarchy), few people will claim you are unreasonably
squeamish.
Whether it makes sense to talk about it or not is the main
question.
If the cumulative hierarchy picture is coherent, then the
question of
whether a set is in it is straightforward. Is it a pure,
well-founded
set? If so, itÕs in the hierarchy; if not, it \
isnÕt.
If the cumulative hierarchy picture is incoherent in some way
that
we donÕt understand yet, then we will have to agree that
weÕve been
looking at the whole situation the wrong way all this time.
If there
were another model, that would ease the pain a little, but
wouldnÕt
mean that we could avoid redoing most of what set theorists
have been
doing.
| But if you could answer that question, then
|the model would have to contain itself,
Why? The cumulative hierarchy does not contain itself. It
contains
only sets. The whole hierarchy is a proper class.
If we restrict to some submodel like the sets of rank less
than the
\first inaccessible cardinal (assuming one exists), then it
doesnÕt
contain itself because it has rank equal to the \first
inaccessible
cardinal.
|and then you could start
|asking self-referential questions, and worse you can able to
make a
|set that was missing. This isnÕt a problem for most
*applications* of
|set theory since you need to consider the whole set
theoretical
|universe to see the failure. ItÕs the whole universe
question again,
|in model-theoretical clothing.
IÕm afraid I donÕt see what the problem is.
|You just always need something bigger to talk about what you
talked
|about before.
Sure.
|But back to implication. If something is truly
|implied, then for any semantically incomplete
|inconsistent-but-isnÕt-strong-enough-to-know-it model, then
it will
|appear to be true, and that appears to be good enough for
most
|mathematicians,
YouÕre personifying the model in a way that makes it sound
like some
real person might be misled in the same way. But thatÕs not
how
mathematics appears to people.
|so proving theorems has a meaning, even if there is
|no model of set theory. It still seems like chasing a
chimera to me,
|IÕm not sure what the point is. WouldnÕt it \
be better to
stick to
|things that DO have models?
Sure. The case against the cumulative hierarchy seems mighty
weak
to me, however.
Keith Ramsay
===
Subject: any smart way of proving |sinc(x)| integrates to
in\finity?
Hi all,
I want to know if there is any smart way of proving
Integrate(|sinc(x)|, from -inf to +inf) = in\finity.
By the way,
does knowing that
Integrate(sinc(x), from -inf to +inf) = 1 and
Integrate(sinc^2(x), from -inf to +inf) = 1
help me solve the problem?
===
Subject: Re: any smart way of proving |sinc(x)| integrates to
in\finity?
>I want to know if there is any smart way of proving
>Integrate(|sinc(x)|, from -inf to +inf) = in\finity.
Since itÕs symetric about the y-axis, you can reduce the
problem to 0 to
+inf if you wish. The zero-lobe is \finite, so you can safely
ignore it.
One tactic you could use is to prove that the area of each
lobe
(x=(2*n-1)*pi/2 to (2*n+1)*pi/2)) is greater than the nth
term in
some series known to be in\finite, say (a_constant)/n.
>does knowing that
>Integrate(sinc(x), from -inf to +inf) = 1 and
>Integrate(sinc^2(x), from -inf to +inf) = 1
>help me solve the problem?
I donÕt think so. And I think you mean sinc(pi*x), at least
in the \first
case.
--Keith Lewis klewis {at} mitre.org
The above may not (yet) represent the opinions of my employer.
===
Subject: Re: any smart way of proving |sinc(x)| integrates to
in\finity?
>I want to know if there is any smart way of proving
>Integrate(|sinc(x)|, from -inf to +inf) = in\finity.
Sure. ItÕs a periodic function, so the integral is equal to
the
integral of one period times the number of periods.
--
Stan Brown, Oak Road Systems, Tompkins County, New York, USA
http://OakRoadSystems.com/
And if youÕre afraid of butter, which many people are nowa-
days, (long pause) you just put in cream. --Julia Child
===
Subject: Re: any smart way of proving |sinc(x)| integrates to
in\finity?
>>Integrate(|sinc(x)|, from -inf to +inf) = in\finity.
> Sure. ItÕs a periodic function, so the integral is equal \
to
the
> integral of one period times the number of periods.
Except itÕs not periodic. sinc(x) = sin(x) / x. It goes up
and down but the
amplitude also tapers off.
===
Subject: Re: any smart way of proving |sinc(x)| integrates to
in\finity?
>Hi all,
>I want to know if there is any smart way of proving
>Integrate(|sinc(x)|, from -inf to +inf) = in\finity.
>By the way,
>does knowing that
>Integrate(sinc(x), from -inf to +inf) = 1 and
>Integrate(sinc^2(x), from -inf to +inf) = 1
>help me solve the problem?
I donÕt think so. Just compute:
Int [kPi..(k+1)Pi] |sinx /x| dx =
Int [0..Pi] |sinu /(kPi + u)| du >=
Int [0..Pi] |sinu /(k+1)Pi)| du =
2/((k+1)Pi).
Now what is known about sum [k=0..oo] 1/(k+1) ?
Thomas
===
Subject: any alternative ways of proving stationarity of
random process or
disproving it?
Hi all,
I have a random sequence, and I have checked
their \first order PDF, f(1), f(2), f(3) ... they are all the
same
distribution...
I have also checked the 2nd order joint distribution,
f(1, 2) and f(2, 3), f(3, 4), ... etc. they are also the same
distribution...
I have also checked E(t) and E(1, 2) and E(2, 3), etc. they
are still the
constant...
I cannot always keep checking these things... right?
Are there any other way of proving stationarity or disproving
it?
===
Subject: Re: any alternative ways of proving stationarity of
random process
or disproving it?
> Hi all,
> I have a random sequence, and I have checked
> their \first order PDF, f(1), f(2), f(3) ... they are all the
same
> distribution...
> I have also checked the 2nd order joint distribution,
> f(1, 2) and f(2, 3), f(3, 4), ... etc. they are also the
same
> distribution...
> I have also checked E(t) and E(1, 2) and E(2, 3), etc. they
are
still
> the constant...
> I cannot always keep checking these things... right?
> Are there any other way of proving stationarity or
disproving it?
It is not clear what you mean by I have a random sequence ...
is
this a sample of a single sequence, samples of many
sequences, or a
theoretical model? If the last, you might be able to do
something by
induction (on the order of the joint distibution being
considered) if
you are after something speci\fic to your model .... otherwise
try to
\fit the model into a class which is known to be stationary or
for
which stationarity tests already exist The most likely
candidates
would be Markov chains or Markov processes, in continuous or
discrete
time. Consider also semi-Markov and recurrent processes as
special
cases of these.
David Jones
===
Subject: Re: Help with Diagonal subgroup problem
>Once you have proven this, try your understanding by proving
the
>generalization known as GoursatÕs Lemma:
>DEF. A subdirect product of two groups H and K is a subgroup
G of
>the direct product H x K, such that the canonical
projections p_1:G->H
>and p_2:G->K are both surjective (that is, for every h in H
there
>exists y in K such that (h,y) in G, and for every k in K
there exists
>x in H such that (x,k) is in G).
>GOURSATÕS LEMMA. Let H and K be two groups. There exist
normal
>subgroup M of H and N of K such that H/M is isomorphic to
K/N if and
>only if there exists a subdirect product G of H and K such
that (G
>intersect H) = M and (G intersect K)=N.
busy here lately.
===
Subject: Normalizers, Centralizers and orbits
I have the following problem:
Let P be a subgroup of S_n (the symmetric group) where P is
of prime order
and
suppose x belongs to S_n normalizes but DOES NOT centralize
P. Show that x
\fixes at most one point in each orbit of P.
[My proof is more of a wordy explanation. I just canÕt think
of how to
prove
this explicitly]
Proof: Let P be a subgroup of S_n of prime order. So
P=(1,.....,p).
Suppose x
belongs to S_n \fixes 1. Now if x \fixes any other \
point in
(1,....,p) and
normalizes (1,2,....,p) then it \fixes all points 1,2,....,p.
Then x would
be
an element of the centralizer. So if x \fixes 2 points in an
orbit of P,
then
it centralizes P. Therefore x \fixes at most one piont in each
orbit of P.
===
Subject: Re: Normalizers, Centralizers and orbits
>I have the following problem:
>Let P be a subgroup of S_n (the symmetric group) where P is
of prime order
and
>suppose x belongs to S_n normalizes but DOES NOT centralize
P. Show that
x
>\fixes at most one point in each orbit of P.
>[My proof is more of a wordy explanation. I just canÕt \
think
of how to
prove
>this explicitly]
>Proof: Let P be a subgroup of S_n of prime order. So
P=(1,.....,p).
Suppose x
>belongs to S_n \fixes 1. Now if x \fixes any other \
point in
(1,....,p) and
>normalizes (1,2,....,p) then it \fixes all points 1,2,....,p.
Then x would
be
>an element of the centralizer. So if x \fixes 2 points in an
orbit of P,
then
>it centralizes P. Therefore x \fixes at most one piont in each
orbit of P.
You need a bit more detail than that. Let P = < g >. Let
{1,...,p} be
an orbit of P where g = (1,2,...,p) ..., and suppose x
normalizes P
and \fixes the two points 1 and t of {1,...,p}. Let h = \
g^(t-1),
so h(1) = t. Then x h x^-1 (1) = x h (1) = x(t) = t. So x h
x^-1 is
a power of h that maps 1 to t, and hence x h x^-1 = h, and x
centralizes
h and hence also g, because P = < h > = < g >.
Derek Holt.
===
Subject: Re: Normalizers, Centralizers and orbits
> I have the following problem:
> Let P be a subgroup of S_n (the symmetric group) where P is
of prime order
and
> suppose x belongs to S_n normalizes but DOES NOT centralize
P.
I understand what you mean here, but you ought to write
something like
... x belongs to S_n and x normalizes .... That would make it
easier to
read.
> ... Show that x
> \fixes at most one point in each orbit of P.
> [My proof is more of a wordy explanation. I just canÕt
think of how to
prove
> this explicitly]
> Proof: Let P be a subgroup of S_n of prime order. So
P=(1,.....,p).
Actually, P could be generated by a product of several
disjoint p-cycles.
Your notation is a bit sloppy, by the way. The subgroup
generated by the
element (1,...,p) is usually denoted <(1,...,p)>.
> ... Suppose x
> belongs to S_n \fixes 1. Now if x \fixes any other \
point in
(1,....,p) and
> normalizes (1,2,....,p) then it \fixes all points 1,2,....,p.
This is the crucial point of the proof. You need to explain
this in more
detail. Also, since the generator of P could be a product of
more than
one p-cycle, you need to think about what you mean when you
say
that x normalizes (1,2,...,p).
> ... Then x would be
> an element of the centralizer. So if x \fixes 2 points in an
orbit of P,
then
> it centralizes P. Therefore x \fixes at most one piont in
each orbit of
P.
YouÕve got the right idea, you just need a bit more detail.
Asger.
-----
Life is wet, then you dry.
===
Subject: Re: Info For Field Theory
posting-account=s2VYhQ0AAADodzR8AJ5syyVwGq_MU-An
I strongly advise the following site :
http://www.jmilne.org/math/index.html
Dusan
===
Subject: New Calculator Words
HereÕs some new words that can be displayed on the old-style
LED or LCD
calculators.
Shit
1.145
Imagine the upsidedown .1 as half a T
BullShit
1.1457778
Imagine pronouncing 3 LÕs as ull
GoToHell
773401.09
Dildo
0.0710
===
Subject: Re: Counterexample to t( (c^n - a^n) mod b ) | phi(b)
>EulerÕs totient theorem gives us b | c^(phi(b)) - 1
>Since c^(n+phi(b))-c^n = c^n*(c^(phi(b)) - 1), the result is
immediate.
Just typing this out as a learning aid, and for future
reference:
EulerÕs totient theorem:
a^phi(n) == 1 mod n
c^phi(b) == 1 mod b
c^phi(b) - 1 == 0 mod b
b | c^phi(b) - 1
b | x * (c^phi(b) -1)
b | c^n * (c^phi(b) -1)
b | c^n*c^phi(b) - c^n
b | c^(n+phi(b)) - c^n
c^(n+phi(b)) - c^n == 0 mod b
c^(n+phi(b)) mod b = c^n mod b
for all n, so c^n mod b has period t | phi(b).
Got it! Not rote learning, but learning by writing, the way
that works for
me.
What is immediate for one is immediate for another after
study.
>> The counter example a,b,c = 5,6,7 shows that the period of
>> c^n - a^n (mod b)
>> does not necessarily divide phi(b).
>ThatÕs not a counterexample, as has already been pointed \
out
to you.
Yes, I \finally got my phi function written right, with help.
You didnÕt
*quite*
have to bang me over the head with a stick, but I did need to
be corrected
three times.
>> We donÕt need to consider the case gcd(b,c)>1.
gcd(a,b,c)=1 is given in
the
>OP to this thread.
>Right. You gave gcd(a,b,c)=1. You did not give
>gcd(a,b)=gcd(a,c)=gcd(b,c)=1. Consider a=6, b=10, c=15. Then
>gcd(b,c)=5 but gcd(a,b,c)=1.
You know, I do need to consider gcd(b,c)>1 because all I am
sure of is
(with my comments interposed) on page 2 that:
If s,y,z are non-zero integers such that x^n + y^n = z^n, if
d=gcd(x,y,z)
and
x1=x/d, y1=y/d, z1=z/d then x1^n+y1^n=z1^n, (which I totally
get) where the
non-zero integers x1, y1, z1 are pairwise relatively prime
(which I donÕt
get).
So if we assume that FermatÕs equation has a non-trivial
solution, it has
one
with pairwise relatively prime integers.
I just donÕt see how factoring out a denominator common to
all *three* of
x,y,
and z leaves x and y, y and z, and x and z with no common
denominator, in
*pairs*.
n,a,b,c = 2,3,4,5 or 2,6,8,10 are solutions to a^n + b^n =
c^n. And IÕve
never
seen a (base?) Pythagorean triangle that had a common factor
between two
edges.
It seems to me x=x1md, y=y1md, z=z1d is possible.
I tolerance everything and tolerate everyone.
I love: Dona, Jeff, Kim, Kimmie, Mom, Neelix, Tasha, and Teri,
alphabetically.
I drive: A double-step Thunderbolt with 657% range.
I \fight terrorism by: Using less gasoline.
===
Subject: Re: Counterexample to t( (c^n - a^n) mod b ) | phi(b)
> If s,y,z are non-zero integers such that x^n + y^n = z^n, if
d=gcd(x,y,z)
> and x1=x/d, y1=y/d, z1=z/d then x1^n+y1^n=z1^n, (which I
totally get)
> where the non-zero integers x1, y1, z1 are pairwise
relatively prime
> (which I donÕt get). So if we assume that \
FermatÕs equation
has a
> non-trivial solution, it has one with pairwise relatively
prime
integers.
> I just donÕt see how factoring out a denominator common to
all *three* of
> x,y, and z leaves x and y, y and z, and x and z with no
common
> denominator, in *pairs*.
> n,a,b,c = 2,3,4,5 or 2,6,8,10 are solutions to a^n + b^n =
c^n. And IÕve
> never seen a (base?) Pythagorean triangle that had a common
factor
between
> two edges.
> It seems to me x=x1md, y=y1md, z=z1d is possible.
In the particular case of x^n + y^n = z^n, any prime which
factors two
of x, y, and z must also factor the third.
For example, suppose p | x and p | y.
Then p | x^n and p | y^n.
Then p | x^n + y^n = z^n.
And one of the properties of prime numbers lets us use p |
z^n to
conclude p | z.
The same works for subtraction as well as addition, in case z
is one of
the two.
--
Daniel W. Johnson
panoptes@iquest.net
http://members.iquest.net/~panoptes/
039 53 36 N / 086 11 55 W
===
Subject: Re: Counterexample to t( (c^n - a^n) mod b ) | phi(b)
>>phi(6)=2 (5 and 1 do not divide 6)
>Neither does 4.
But it doesnÕt matter. Totatives of n are coprime to n
whether they divide n
or
not.
phi(6)=2. No counterexample.
And so, we have
t((a^n+b^n) mod c) | phi(c)
t((c^n -a^n) mod b) | phi(b)
t((c^n -b^n) mod a) | phi(a)
and my apology for writing bad code.
language in Mathcad.
Got it now, for keeps.
I tolerance everything and tolerate everyone.
I love: Dona, Jeff, Kim, Kimmie, Mom, Neelix, Tasha, and Teri,
alphabetically.
I drive: A double-step Thunderbolt with 657% range.
I \fight terrorism by: Using less gasoline.
===
Subject: Orthonormal basis of an in\finite-dimensional Hilbert
space
How do I prove that the basis is NOT countable if I take only
FINITE
linear combinations of vectors from the basis?
*-----------------------*
www.GroupSrv.com
*-----------------------*
===
Subject: Re: Orthonormal basis of an in\finite-dimensional
Hilbert space
> How do I prove that the basis is NOT countable if I take
only FINITE
> linear combinations of vectors from the basis?
Suppose {v_1, v_2, ...} is a (Hamel) orthogonal basis of an
inner space H.
We show that H cannot be complete by constructing a
Cauchy-sequence that
does not converge.
After renormalization, it can be achieved that ||v_n|| =
2^(-n). Now
consider the sequence x_n = v_1 + ... + v_n. Because of the
triangle
inequality and the geometric series, this is a Cauchy
sequence. Further
suppose that its limit is y = y_1*v_1 + ... + y_m*v_m
(because the v_i are
a Hamel basis, such a m exists). But, for n > m,
|| x_n - y || >= 2^(-m-1) + 2^(-m-2) + ...
which shows that the sequence cannot converge to y.
Hm, at \first I thought that somehow one could generalize this
argument to a
normed space. Does anyone see how?
--
everyone who casts a shadow
seems to stand in the sun
reverse my forename for mail! - saibot
===
Subject: re:Orthonormal basis of an in\finite-dimensional
Hilbert space
I know that BaireÕs theorem is needed.. I just \
donÕt know how
to use
it here.
*-----------------------*
www.GroupSrv.com
*-----------------------*
===
Subject: Re: Orthonormal basis of an in\finite-dimensional
Hilbert space
> I know that BaireÕs theorem is needed.. I just \
donÕt know
how to use
> it here.
Hint:
Any \finite-dimensional subspace is closed and nowhere dense.
--
G. A. Edgar
http://www.math.ohio-state.edu/~edgar/