mm-175
===
Subject: Clarication with announcement of new
version
Let me answer your
question:>Hello ! Now, these points might have been addressed
before but I would>really>appreciate some clarications in
your proof :>>- In Def. 2.2, what does it mean .95we 
dene that
each covering has at>least two>members' ? Does it simply 
mean
that, if J has got one or no member we>do not>call it a
covering ?For the sake of clarity, I declare:We shall not call
what you would call a covering with one point asa covering.
Thus, if you dare to argue with that argument, sayingthat a
covering with one point is nonsesne, will be treated
asnonsense, since there does not exist any such a thing in my
proof.>>- In Def. 2.3, what is the meaning of indexing the
left subscript of>J' ? That>is, what is the meaning of r_i
?That thing is for residue. >You explain that p_i is the i-th
odd>prime,>but it is unclear what the subscript 
.95i' means on
r.For detail, please refer to the new version. It relates to
therestriction on the distinctness of prime moduli used.>-
Same one, regarding the denition of P' as an 
intersection of
P and>N. You appear to have dened things such that P(3,pl) 
is
a subset of>N(3,q)_odd (since pl <= q), so why take the
intersection or introduce>P' for that matter ?This is to let
my mathematical induction happen. If we do notdene a
universal set, then the whole thing of For every q ...becomes
nonsense.>- A bit later, in the denition of A <subset> B, 
you
appear to be>putting as conditions the equivalence of A and B
simultaneously with>the existence of unpaired points of B !
Doesn't the equivalence imply>the nonexistence of such 
points
?Doesn't the equivalence imply the nonexistence of such
points? ???a b c d e =: Ae f g i j =: BSay A is equivalent to
B. With a corresponding positionwisely to e.Because they are
equivalent, there exist pair of points. Did I getyour question
correctly?>At this point I stopped reading. Could you please
clarify or correct>any misunderstanding of mine ?Please read
on. Check out Nature for Dr. Perelman. ( I would be sorryfor
Dr. Perelman on making his name public, but he would be
quiteinspiring to me as Paul Erdos would be to me. ) Just
please try tounderstand with scrutiny. I am almost giving up
publishing the result.But amazingly, Dr. Perelman has also not
shown an interest ofpublishing his result. (Perhaps because of
:)Hisanobu ShinyaNew version will be available
inhttp://www.mathpreprints.comor my geocities
website.
approve@localhost) by support1.mathforum.org
(8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id
sci.mathi don't know how to post without it being a 
response,
Unit Circle, and i found images that only had 3 points in each
Quadrant. why don't they show things like pi/12, 5pi/12, 
7pi/12
...23pi/12 and the point it lies on, on the unit circle. i have
tofind out what the cosine of 19pi/12 for homework, but i 
only
come up with an approximate value, not the exact one. i only
ask for 8 points on the unit circle, can someone help
me?
===
Subject: Re: limited unit circle> i have tofind out what
the cosine of 19pi/12 for homework, but i > only come up with
an approximate value, not the exact one. i only ask > for 8
points on the unit circle, can someone help me?Would it help
you any if I were to point out that 19/12 = 15/12 + 4/12?--
Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for
email)
Myerson <gerry@maths.mq.edi.ai.i2u4email>,
cosine of 19pi/12 for homework, but i >> only come up with an
approximate value, not the exact one. i only ask >> for 8
points on the unit circle, can someone help me?>>Would it help
you any if I were to point out that >19/12 = 15/12 + 4/12>?Or
as I had done, 19/12 = 16/12 + 3/12.Rob Johnson
<rob@trash.whim.org>take out the trash before
replying
===
Subject: Re: Candy Inspiration (in the news)> M & M
packing:
index.html I guess I am not surprised they pack better than
spheres. If M&Ms had been VERY ¤at, it seems intuitively
obvious they would> have packed even more efciently.> (as
long they were oriented in the barrel so that neighboring
oblate> spheroids were mostly positioned, on average, in
somewhat the same> direction.)It seems intuitive to me too. If
you imagine M&M's being arbitrarily long and thin 
(topological
M&Ms) you can see that they approach the packing density 
you'd
get if all the M&M's were ¤at -- namely 100%. 
I'd guess that
spherical M&Ms are the LEAST efcient shape for
packing.
===
Subject: Re: Candy Inspiration (in the news)shfry
 If M&Ms had
been VERY ¤at, it seems intuitively obvious they would|> have
packed even more efciently.|> (as long they were oriented in
the barrel so that neighboring oblate|> spheroids were mostly
positioned, on average, in somewhat the same|> direction.)||It
seems intuitive to me too. If you imagine M&M's being
arbitrarily |long and thin (topological M&Ms) you can see that
they approach the |packing density you'd get if all the 
M&M's
were ¤at -- namely 100%. I'd |guess that spherical M&Ms 
are
the LEAST efcient shape for packing.I can believe they might
pack better, but I don't see why. If they are allellipsoids
oriented the same way, i.e. with their three corresponding
axesparallel, it's impossible to get a better packing 
density
than what youcan get for spheres, because there is a linear
transformation whichtakes all of them from ellipsoids to
spheres.In your argument, you have two limits being taken. One
is the limit asthe shape becomes very ¤at. The other is the
limit in the denition ofpacking density, namely as the 
volume
of the box goes to innity.I don't see how you 
can get a
packing density close to 100% for anygiven (ellipsoidal)
shape, given that they are thicker in the middle andthinner at
the edges. If you stacked them up like pancakes, for
example,you'd still have that wasted space at the edges, 
plus
the fact that there'sno way to ll the plane 
with disks
without gaps.into big containers, which surely is less than
the ideal best possibledensity. I can believe that M&Ms might
be better at packing together insomething like the optimal
spherical packing (with the linear transformationapplied to
turn them into ellipsoids instead of spheres) than spheres
are.It would still be interesting to know, though, whether
there's any shape ofellipsoid whose *best* packing density
exceeds the *best* density forspheres.Keith Ramsay
===
Subject:
constructing realsHi all,i had this idea of dening the reals
from rationals by intersectionsofsubsets of Q:Let {O_n} be a
sequence of open intervals in Q s.t. |O_n| -> 0and O_n+1 is in
O_n for all n.Then either U(O_n)=p/q is in Q, or U(O_n)={} in
which case U(O_n)denes an irrational number.Would this be
only slightly different from the approach through
comments,typek
===
Subject: Re: constructing reals> Hi all, i
had this idea of dening the reals from rationals by
intersections> of> subsets of Q:> Let {O_n} be a sequence of
open intervals in Q s.t. |O_n| -> 0> and O_n+1 is in O_n for
all n.> Then either U(O_n)=p/q is in Q, or U(O_n)={} in which
case U(O_n)> denes an irrational number.> Would this be only
slightly different from the approach through the> equivalence
classes of cauchy sequences ?Given such a sequence of open
intervals, a suitable Cauchy sequence canbe easily constructed
by interleaving the endpoints. Alternately, thesequence of
lower endpoints and the sequence of upper endpoints
wouldthemselves be suitable Cauchy sequences. Constructing a
nested sequenceof open intervals from a Cauchy sequence is not
always that easy, but Ithink it is always possible (although
some of the endpoints might not beterms in the sequence).I
think using intervals like this is an unneeded complication.
-- Daniel W.
Johnsonpanoptes@iquest.nethttp://members.iquest.net/~panoptes/
039 53 36 N / 086 11 55 W
===
Subject: Re: constructing
> .... > i had this idea of
dening the reals from rationals by intersections> of> 
subsets
of Q:> Let {O_n} be a sequence of open intervals in Q s.t.
|O_n| -> 0> and O_n+1 is in O_n for all n.> Then either
U(O_n)=p/q is in Q, or U(O_n)={} in which case U(O_n)> denes
an irrational number.... See Richard Courant & Herbert
Robbins, What is Mathematics? pp.68-71, General Denition of
Irrational Numbers by Nested Intervals. They've got a nice
diagram too. Ken Pledger.
===
Subject: the anticlassicalist }{
i: linguistic negation-=-=-=-=-= linguistic negation
=-=-=-=-=-=-The logic of natural language has been studied by
many different schools ofthought throughout history. Much of
this study has pointed to many modelsdifferent than those
based on classical logical structures. I have studiedthis 
eld
for some time, but I have had some questions along the way.I
was looking at litotes (perhaps more exactly, antenantiosis),
with formslikeHe is not unattractive.It is not impossible to
do that.where the double-negation implies some form of a shift
in meaning foreffect, following closely an empirical maxim
sometimes called the divisionof pragmatic labor:The use of a
longer, marked expression in lieu of a shorter
expressioninvolving less efort on the part of the speaker
tends to signal that thespeaker was not in a position to
employ the simpler version felicitously(L. R. Horn, Duplex
negatio afrmat... the economy of double negatives -- Chicago
Linguistic Society 27-II)This kind of maxim, though, is very
general, and would seem to point to manyof the other
non-classically negated logics which still support the
partialorder implied. This shows some sort of felicity with
the agrammatical butcommon use of double negatives to stress
negativity, such asThey don't know nothing!In the logical
literature I have read, unfortunately, there is not as
muchdistribution analyses of these logical models (for
instance, thePerspectives der Analytischen Philosophie book
Negation: a notion in focusedited by Heinrich Wansing), and so
I was hoping someone might hold someclues as to:1) Are there
any good tables available in the literature that
correlatestypes and uses of negation with modern and
historical languages? I mean, Iknow of some specic works 
such
as The syntax of negation in russian bySue Brown, Buddhist
theory of meaning (apoba) and negative statement byDhirenda
Sharma, The grammar of negation by Jong-Bok Kim, and have
readthings about Serbian and similar forms, but I'm very
unsure of what isconsidered the most modern or agreed upon
population correlations withfairly comprehensive
scope.Additionally, I have been intrigued by the denial of
terms found in theanvitabhidhana school, where negation is
used to give a meaningful directionto action. For example, one
may sayThe not-house will be builtTo stress the otherness or
change for which the building is applied (as adistinguishing
in the existing forms). This richness is an earlycontribution
towards a typed theory of semantics. Su (la ilaha 
illa'llah)
and taoist theories of negation are also quite sophisticated,
and soI was curious about:2) are there also well known schools
of thought about negation from Africanor South American
traditions?--
-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-galathaea:
===
prankster, fablist, magician, liarSubject: Re: the
anticlassicalist }{ i: linguistic negation
galathaea
expression in lieu of a shorter expression : involving less
efort on the part of the speaker tends to signal that the :
speaker was not in a position to employ the simpler version
felicitously : (L. R. Horn, Duplex negatio afrmat... the
economy of double negatives -- : Chicago Linguistic Society
27-II) : : This kind of maxim, though, is very general, and
would seem to point to many : of the other non-classically
negated logics which still support the partial : order
implied. This shows some sort of felicity with the
agrammatical but : common use of double negatives to stress
negativity, such as : : They don't know nothing!This is 
simply
not true.This is not really a double negative.Compare with they
don't know jack,they don't know diddly, orthey 
don't know
squat.These are not double negatives and they all mean thesame
thing.There is a relevant ellipsis here.If you include what is
being left outthen all these are grammatical.If you go 49.9
miles then you have not gone50 miles. If there are 50 relevant
facts andyou only know 49 of them then you don't know all
50.Suppose that instead of being merely relevant, thesefacts
are basic. Then it could be truthfully said aboutthem (if they
don't know All these facts) that they don't 
know the basics.The
point in all these cases is that the ellipsis isabout some such
phrase as as much as or even as muchas. The deep structure
under the surface structure ofthey don't know x is they 
don't
know an amount AS MUCH AS amount x,i.e., they know EVEN LESS
THAN amount x.If I ask you for money and you reply, I won't
give you a dime,you are not implying that there is some
non-zero chance thatyou will give me a thousand dollars. What
this rather means isI won't give you EVEN AS MUCH AS a dime,
or equivalently,WhatEVER I give you, it will be LESS than a
dime.They don't know *nothing*! (the exclamation pointDOES
make a difference; it implies that my emphatic*'s around
nothing were originally intended) is just a more intense
version of all these other single-negativesinvolving a
positive-yet-smaller-than-epsilon quantity.It is ultimate. In
some cases there really is a differencebetween supplying a
zero amount of something and not showingup to supply anything
at all (i.e. giving somebody an emptybag and not even botherin
to show up with a bag).RATHER than being a double negative,
They don't know nothingREALLY means that they 
don't know even
AS MUCH as nothing,that they know LESS than nothing, or that
even allegingabout them that they MIGHT know an amount EVEN As
small as 0is inappropriate; they are just not the kind of
people towhom knowing IN GENERAL even COULD be
ascribed.
===
Subject: Re: the anticlassicalist }{ i: linguistic
negation -=-=-=-=-= linguistic negation =-=-=-=-=-=- The logic
of natural language has been studied by many different schools
avoided coming to any unique conclusionemphasizes the
bankruptcy of the Liberal Arts. It's a hen party. Anybody 
can
gossip and all are appreciated. It's bull. How wouldyou
like to be doing 80 mph in a car that was built by mob
ruleinstead of cold, hard, dry engineering?Oh, but that
example is different! It certainly is. Liberal Arts
arebull and engineering is real world.Everything in the
Liberal Arts and Social Sciences is both right andwrong. No
conclusion at all can be drawn except the need for
morefunding. Say what you want - it makes no difference at all
(exceptfor grant funding - then you must agree with the mob or
be destroyed).Klingon was specically created to be the worst
language possible byfolks who knew linguistics. It was
enthusiastically embraced by themob and it is as good as
Korean or Chinese for transferring content.--Uncle
Alhttp://www.mazepath.com/uncleal/qz.pdfhttp://
www.mazepath.com/uncleal/eotvos.htm (Do something naughty to
physics)
===
Subject: Re: the anticlassicalist }{ i: linguistic
negation: >: > -=-=-=-=-= linguistic negation =-=-=-=-=-=-: >:
> The logic of natural language has been studied by many
That they have one and all avoided coming to any unique
conclusion: emphasizes the bankruptcy of the Liberal Arts.
It's a hen party.: Anybody can gossip and all are 
appreciated.
It's bull. How would: you like to be doing 80 mph in a 
car
that was built by mob rule: instead of cold, hard, dry
engineering?:: Oh, but that example is different! It certainly
is. Liberal Arts are: bull and engineering is real
world.Formal linguistics is founded on the same rigour that
onefinds in quantummechanics, so don't give me 
your bull.
Obviously, I wouldn't trust alinguist to build my car solely
on linguistic credentials, just as Iwouldn't trust a 
physicist
to do surgery on me solely on the physicscredentials.in any way
condemning science. I am scientist, and maybe if you stop
yourre¤ex vomit and read a little, maybe one day you will
become one too.I am talking about what we mean by making
propositions in a model, and willbe discussing all sorts of
wonders, like quantum logic, causal sets, andeven maybe
explore antigen-antibody reactions. Yes, I will also be
givingmany linguistic examples. I am sorry that I won't be
able to respect thenarrow mindedness that you are so
comfortable with.: Everything in the Liberal Arts and Social
Sciences is both right and: wrong. No conclusion at all can be
drawn except the need for more: funding. Say what you want - it
makes no difference at all (except: for grant funding - then
you must agree with the mob or be destroyed).:: Klingon was
specically created to be the worst language possible by:
folks who knew linguistics. It was enthusiastically embraced
by the: mob and it is as good as Korean or Chinese for
transferring content.Awww... I was hoping for a little more
blatant racism from you. I can'thave any fun tearing you a 
new
one with that tired rag.--
-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-galathaea:
prankster, fablist, magician, liar
===
Subject: Re: the
anticlassicalist }{ i: linguistic negation : >> : > -=-=-=-=-=
linguistic negation =-=-=-=-=-=-> : >> : > The logic of natural
language has been studied by many different schools> of> : >
and all avoided coming to any unique conclusion> : emphasizes
the bankruptcy of the Liberal Arts. It's a hen party.> :
Anybody can gossip and all are appreciated. It's bull. 
How
would> : you like to be doing 80 mph in a car that was built by
mob rule> : instead of cold, hard, dry engineering?> :> : Oh,
but that example is different! It certainly is. Liberal Arts
are> : bull and engineering is real world. Formal
linguistics is founded on the same rigour that onefinds in
quantum> mechanics, so don't give me your bull.
has pulled adonut about itself. It proposes to follow
scientic method ondatabases that are at best weakly
statistical and that have noempirical falsication possible,
inputs or outputs. Like psychology,it shouts its
self-percieved triumphs and whimpersheteroskedasticity at its
failures. BTW, what are linguistics'triumphs?Well I 
don't
think it's quite fair to condemn a whole program becauseof a
single slip-up, sir. General Buck Turgidson explaining to
thePresident the psychological evaluation program whose
slip-up wasGeneral Ripper's 34-plane airborne nuclear attack
upon Russia. --Uncle
Alhttp://www.mazepath.com/uncleal/qz.pdfhttp://
www.mazepath.com/uncleal/eotvos.htm (Do something naughty to
physics)
===
Subject: Re: the anticlassicalist }{ i: linguistic
negation: > Formal linguistics is founded on the same rigour:
> that onefinds in quantum mechanics, so don't: 
> give me your
hole that has pulled a: donut about itself. It proposes to
follow scientic method on: databases that are at best weakly
statistical and that have no: empirical falsication 
possible,
inputs or outputs. Like psychology,: it shouts its
self-percieved triumphs and whimpers: heteroskedasticity at
its failures. BTW, what are linguistics': triumphs?Well, the
rst and most apparent of all triumphs has been the ability
toformalise and rigorise communication to avoid the ambiguity
of everyday talkand create that useful thing called science.
Logical linguistics andsemiotics has abstracted in many
different directions, including describingthe foundations of
mathematics and the logical structure of scienticmodels. It
is also able to describe a lot of structure in natural
language,but I don't want to distract you.There is even a 
lot
of research mapping neural connections, crammingelectrodes
into the brains of various eumetazoan critters,
determiningglucose consumption through radiological methods,
or nicely detailing RNAexpression or neurotransmitter response
to freshly sliced brain sections ofvarious experimental
subjects. You can block your eyes and plug your earsall you
want, but there is already a quite large and developing
communitystudying the origin of auditory and visual symbolic
processing in the brain,and strangely enough, we already have
the beginnings of linguisticexpression forming in the
research. And they match quite nicely withlogical linguistic
research that formal languages are built on.So where do you
cut the line, mister scientist? Where do models stop,through
whatever mystical explanation you care to provide, being able
toaccurately represent reality? Have you not seen the quite
accurate modelsthat ll research proceedings every year on
modeling various brain regions?You could even use your amazing
snipped all the references to math and physics,
perhapsindicative that you never wanted to engage in
meaningful dialectic? That istypical for one such as yourself
who considers their daily minesweeperpractise innitely more
useful to society than the farm workers who'scitizenship may
be questioned.--
-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-galathaea:
===
prankster, fablist, magician, liarSubject: Re: the
anticlassicalist }{ i: linguistic negation
> BTW, what are
linguistics'> triumphs?You are my good Sir, and so is the
Universe and everything,even 42. Allow me to prove it... ,
that's the Bookof Mormon... Urantia Book? no.. where are my
Gideons' Bibles when I need them? This will have to
do:L'.8evangile selon Saint JeanAu commencement le Verbe
.8etait et le Verbe .8etaitavec Dieu et le Verbe .8etait Dieu.
Il .8etait aucommencement avec Dieu. Tout fut par lui etsans
lui rien ne fut.Voil.88. On ne saurait .90tre plus clair: in
the beginning was the Word and the Word was withGod and the
Word was God. The Word was in thebeginning with God, and by
Him/It all was made,and without Him/It nothing. (or something
tothat effect).There, Sir, lies the Triumph of
Linguistics.Hors la linguistique, point de salut, c'est
clair.Beat that!
===
Subject: Re: the anticlassicalist }{ i:
linguistic negationgalathaea > The logic of natural language
has been studied> by many different schools of thought
throughout history.> Much of this study has pointed to many
models> different than those based on classical logical
structures.> I have studied this eld for some time, but I
have had some> questions along the way.>> I was looking at
litotes (perhaps more exactly, antenantiosis),> with forms
like>> He is not unattractive.> It is not impossible to do
that.>> where the double-negation implies some form of a
shift> in meaning for effect, following closely an empirical
maxim> sometimes called the division of pragmatic labor:>> The
use of a longer, marked expression in lieu of a shorter>
expression involving less efort on the part of the speaker
tends> to signal that the speaker was not in a position to
employ the> simpler version felicitously> (L. R. Horn, Duplex
negatio afrmat... the economy of double> negatives -- >
Chicago Linguistic Society 27-II)>> This kind of maxim,
though, is very general, and would seem> to point to many of
the other non-classically negated logics> which still support
the partial order implied. This shows some> sort of felicity
with the agrammatical but common use of> double negatives to
stress negativity, such as>> They don't know nothing!>> In 
the
logical literature I have read, unfortunately, there is> not as
much distribution analyses of these logical models> (for
instance, the Perspectives der Analytischen Philosophie> book
Negation: a notion in focus edited by Heinrich> Wansing), and
so I was hoping someone might hold some> clues as to:>> 1) Are
there any good tables available in the literature> that
correlates types and uses of negation with modern> and
historical languages? I mean, I know of some specic> works
such as The syntax of negation in russian by> Sue Brown,
Buddhist theory of meaning (apoba) and> negative statement by
Dhirenda Sharma, The grammar> of negation by Jong-Bok Kim, and
have read things> about Serbian and similar forms, but I'm 
very
unsure> of what is considered the most modern or agreed upon>
population correlations with fairly comprehensive scope.>>
Additionally, I have been intrigued by the denial of terms>
found in the anvitabhidhana school, where negation> is used to
give a meaningful direction to action.> For example, one may
say>> The not-house will be built>> To stress the otherness or
change for which the building> is applied (as a distinguishing
in the existing forms).> This richness is an early
contribution towards a typed> theory of semantics. Su (la
ilaha illa .95llah) and taoist> theories of negation are also
quite sophisticated, and> so I was curious about:>> 2) are
there also well known schools of thought> about negation from
African or South American>
traditions?---------------------------------------------------
--------------They speak mostly Spanish in South America.In
Spanish, double-negatives are required.It has nothing to do
with logic. It's just amatter of grammatical
agreement:http://spanish.about.com/library/weekly/
aa061101a.htm~I don't know if legalese-Spanish is any
different.~People frequently make positive assertions,such as
this is really great..., while at thesame time shaking their
heads as if denying it.When I rst started noticing that, I
couldn'tthink of any explanation for it other thanthat it 
was
probably a parapraxix, like a slipof the tongue, -revealing
that they were lying.But that can't possibly be it.Whatever
the true explanation, I think ithas something to do with how
we constructsentences unconsciously. I think we constructin
parts, in parallel, many different ways to saya thing, and
then, at the last second,choose one of them to speak. And I
thinkwe always construct both positive andnegative ways to say
it. And then, if wechoose to speak the positive form, ourheads
may shake to negate, -not it, - butthe lingering unspoken
(.95undischarged'perhaps) negative form.I believe 
negation of
sentences wasone of the things Chomsky had in mindas a
surface-structure transformation,-meaning that it is a trivial
last-secondtransformation made on a deep-structureconstruct.
Which would only make senseif the deep construct was
un-ambivalent,- perhaps even always positive. But Ibelieve
that Chomsky eventually gaveup that approach.~Different
cultures use differenthead motions to signal 
.95yes' and .95no'.In
India they bob their heads tosay .95yes' -in a way 
that looks
toothers like .95no'. Italians jerktheir head up 
once to say
.95yes',- in a way that looks to otherslike they 
want to start a
ght.(- Some may recall images ofMussolini 
.95chilling' -
crossinghis arms and jerking his headup after giving a
speech;He wasn't looking for a ght;He was just 
pleased with
himself.)Desmond Morris talks aboutgesture frontiers, or
isoclines.And isoclines of yes/no gesturescross each other in
the Balkans.That, and the suspicion that body-languagethat
appears to contradict the spoke languageimplies that the
person is lying - may goa long way to account for the
Balkans.No one understands any one there.~~Sartre regarded
negation as the primarycharacteristic of being human: ....
from a web page: -- On Part One of Being and Nothingness:
Nothingness and Bad
Faithhttp://216.239.51.104/search?q=cache:Vp5XSEbyWXgJ:
web.ics.purdue.edu/~mmichau/sartre-bn.pdf+%22Being+And+
Nothingness%22+negation&hl=en&ie=UTF-8 I. The Origin of
Negation In this section, Sartre discusses the relation
between being-in-itself and being-for-itself. Probably the
most original feature in Sartre's conception of 
consciousness
is his insistence on its essential negativity; neither Husserl
nor Heidegger gives the negative function such a central place
ii. In BN we begin with a sober analysis of investigation as
called for by the ontological problem, similar to the
discussion of Being in Being and Time. However, in Sartre's
work, the main emphasis is on the readiness to be faced by the
non-existence of the situation inquired about. The question
thus reveals that nothings are constant possibilities of our
experience. In fact consciousness is shot through with
nothings; Nothingness is an immanent characteristic of
Being.etc.~~> It is not impossible to do that. ...> The use of
a longer, marked expression in lieu of a shorter> expression
involving less effort on the part of the speaker tends> to
signal that the speaker was not in a position to employ the>
simpler version felicitouslyI don't buy less effort 
playingany
role at all in natural speech.People think at a xed rate. 
And
they also speak atsome xed rate appropriate to circumstance.
Normally,speech is massively redundant and contentless.
Speechis, I think, just a vestigial behavior originating in
ourmonkey ancestor's primary social bonding behaviorcalled
lice picking. We don't carry lice anymore,or 
don't admit it,
and, therefore we have to talk toeach other instead. And so,
there's just no pointinfinding ways to say 
things with .95less
effort'.There's certainly a connotational 
difference between:
It is possibleand It is not impossible.However, it's not
perfectly clearto me what it is, at this moment.~> So I still
am left asking:> 3) what is the actual source of the bias
against nonclassical logics?Try: Deviant Logic, Fuzzy Logic:
Beyond the Formalism -- by Susan
Haackhttp://www.amazon.com/exec/obidos/tg/detail/-/0226311341/
qid=1077019274/sr=1-1/ref=sr_1_1/102-7545475-4339367?v=glance&
s=books~Greg.
===
Subject: Re: the anticlassicalist }{ i:
linguistic negation:: They speak mostly Spanish in South
America.:: In Spanish, double-negatives are required.: It has
nothing to do with logic. It's just a: matter of grammatical
agreement::
http://spanish.about.com/library/weekly/aa061101a.htm: ~:: I
don't know if legalese-Spanish is any different.: ~This
distinction between negation of objects and negation of
actions seemsto have lead to a rich variety of negations in
various languages. I don'tknow why I didn't 
mention Spanish,
but I hear this all the time and make theappropriate
conclusion quite readily. I think this may be common
inPortuguese as well, as I have a Brazillian friend who uses
it quite often(though the language is so pretty, its sometimes
hard to hear the languagefor the song).(though, I will have to
check again, since I really was looking for it). Itis much
more formal and strives to be clear.: People frequently make
positive assertions,: such as this is really great..., while
at the: same time shaking their heads as if denying it.:: When
I rst started noticing that, I couldn't: think 
of any
explanation for it other than: that it was probably a
parapraxix, like a slip: of the tongue, -revealing that they
were lying.:: But that can't possibly be it.Sure it can! But
it seems it could also be an indication of an internalstruggle
elicited by the surprise, as if the internal model needs
updatingbecause of the inconsistency. Which you..: Whatever
the true explanation, I think it: has something to do with how
we construct: sentences unconsciously. I think we construct: in
parts, in parallel, many different ways to say: a thing, and
then, at the last second,: choose one of them to speak. And I
think: we always construct both positive and: negative ways to
say it. And then, if we: choose to speak the positive form,
our: heads may shake to negate, -not it, - but: the lingering
unspoken (.95undischarged': perhaps) negative 
form.:: I believe
negation of sentences was: one of the things Chomsky had in
mind: as a surface-structure transformation,: -meaning that it
is a trivial last-second: transformation made on a
deep-structure: construct. Which would only make sense: if the
deep construct was un-ambivalent,: - perhaps even always
positive. But I: believe that Chomsky eventually gave: up that
approach.:: ~... point out yourself, in a way. There is
certainly a complex selectiveprocess in the construction of
our communication. Negation may be a trivialtransformation
sometimes, but I believe there are many situations where
itsintent is more subtle and must be inferred from other clues
of theconversation, perhaps nonverbal ones like you mention.:
Different cultures use different: head motions to signal 
.95yes'
and .95no'.: In India they bob their heads to: say 
.95yes' -in a
way that looks to: others like .95no'. Italians 
jerk: their head
up once to say .95yes',: - in a way that looks to 
others: like
they want to start a ght.: (- Some may recall images of:
Mussolini .95chilling' - crossing: his arms and 
jerking his
head: up after giving a speech;: He wasn't looking for a
ght;: He was just pleased with himself.):: Desmond Morris
talks about: gesture frontiers, or isoclines.: And isoclines
of yes/no gestures: cross each other in the Balkans.:: That,
and the suspicion that body-language: that appears to
contradict the spoke language: implies that the person is
lying - may go: a long way to account for the Balkans.: No one
understands any one there.:: ~~Ahh, the balkanisation of
negation! Such a combative view of nonclassicaltheories of
negation...With so much to worry about in utterances and
visual symbolics, gesturevariation seems so much more...
easily acceptable? Although I do like tolook at faces closely,
Ifind I rarely see gestures as anything offensivewhen I
understand the speak. This seems a common trade-off
ininterpreting communication, though you bring up powerful
points about realintent often found in the unspoken.:: Sartre
regarded negation as the primary: characteristic of being
human: ....:: from a web page: --: On Part One of Being and
Nothingness:: Nothingness and Bad
Faith:http://216.239.51.104/search?q=cache:Vp5XSEbyWXgJ:
web.ics.purdue.edu/~mmichau/sartre-bn.pdf+%22Being+And+
Nothingness%22+negation&hl=en&ie=UTF-8:: I. The Origin of
Negation: In this section, Sartre discusses the relation:
between being-in-itself and being-for-itself.: Probably the
most original feature in Sartre's: conception of 
consciousness
is his insistence: on its essential negativity; neither Husserl
nor: Heidegger gives the negative function such a: central
place ii. In BN we begin with a sober: analysis of
investigation as called for by the: ontological problem,
similar to the discussion: of Being in Being and Time.
However, in Sartre's: work, the main emphasis is on the
readiness to be: faced by the non-existence of the situation
inquired: about. The question thus reveals that nothings are:
constant possibilities of our experience. In fact:
consciousness is shot through with nothings;: Nothingness is
an immanent characteristic: of Being.::: etc.: ~~I wasn't 
sure
whether or not to bring up Sartre, and I regret that. Ishould
have been more eager to include his work in the schools of
negation Iam studying. I keep letting the biases of some
so-called scientists on theboards affect the focus of my
this up.Negation as primary in cognitive notions of existence
against nothingnessdoes seem to have some type of primary
place in our conceptualisationhidden negations used in
identication processes. I also believe that thisline of
reasoning describes how even rst-order logics rely
uponsecond-order notions implicitly passed off to the user,
and I should tie all: > It is not impossible to do that. ...:
> The use of a longer, marked expression in lieu of a shorter:
> expression involving less effort on the part of the speaker
tends: > to signal that the speaker was not in a position to
employ the: > simpler version felicitously::: I don't buy 
less
effort playing: any role at all in natural speech.:: People
think at a xed rate. And they also speak at: some 
xed rate
appropriate to circumstance. Normally,: speech is massively
redundant and contentless. Speech: is, I think, just a
vestigial behavior originating in our: monkey ancestor's
primary social bonding behavior: called lice picking. We 
don't
carry lice anymore,: or don't admit it, and, therefore we 
have
to talk to: each other instead. And so, there's just no 
point:
infinding ways to say things with .95less 
effort'.:: There's
certainly a connotational difference between::: It is
possible:: and:: It is not impossible.:: However, it's not
perfectly clear: to me what it is, at this moment.: ~I think
the effort is found in the moving of the decision to one that
isconscious. The example of Spanish shows that negative
elements are added ina negative scope quite easily by those
who habituate the negative form. Butthere are certain
expressions that are not common, and are clear in acontext to
have been chosen for a reason. I wouldn't put they 
don't
knownothing in that group, but I would usually put it is not
impossibleunless I was speaking to someone who has completely
habituated suchambiguity defensively.Whether an afrmation or
a negative seems more ¤uid or subconscious in agiven context
is certainly difcult to objectively test, but I do
believethere are indicators found in patterns of speech which
help us judge ifsomething was thought about more deeply before
being uttered.:: > So I still am left asking:: > 3) what is the
actual source of the bias against nonclassical logics?::: Try::
Deviant Logic, Fuzzy Logic: Beyond the Formalism: -- by Susan
Haack::http://www.amazon.com/exec/obidos/tg/detail/-/
0226311341/qid=1077019274/sr=1-1/ref=sr_1_1/102-7545475-
4339367?v=glance&s=booksI had that book in my hands about
three weeks ago, but decided not to checkit out in favor of
some other readings. My local university library has
twocopies, so I should be able to get that later on today when
very helpful.--
-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-galathaea:
prankster, fablist, magician, liar
===
Subject: Re: the
anticlassicalist }{ i: linguistic negation~greg >> ...>
Whatever the true explanation, I think it> has something to do
with how we construct> sentences unconsciously.Meant
.95subconsciously'.(I write posts 
.95unconsciously'.)
===
Subject:
Re: the anticlassicalist }{ i: linguistic negationNow really,
Ms.(?) fabulist, etc: there is much that is more
grosslyinappropriate to sci.physics here, so you might say all
the other kidsare doing it. But if you affect to be a
_superior_ kid, might you notlead by example and limit the
number of freshly cross-posted threadswhich are mainly about
linguistics, and neither physics nor math?You are beginning to
speak Crankish.> -=-=-=-=-= linguistic negation =-=-=-=-=-=-
The logic of natural language has been studied by many
different schools of> thought throughout history. Textual
analysis: schools of thought do not study anything, and is
acliche.For somebody who is so academically concerned with
language, you seemremarkably insensitive to its use. Your
earnestly learned turgiditywill carry you far ... away from
here.<snip>
===
Subject: Re: the anticlassicalist }{ i:
linguistic negation:: Now really, Ms.(?) fabulist, etc: there
is much that is more grossly: inappropriate to sci.physics
here, so you might say all the other kids: are doing it. But
if you affect to be a _superior_ kid, might you not: lead by
example and limit the number of freshly cross-posted threads:
which are mainly about linguistics, and neither physics nor
math?:: You are beginning to speak Crankish.There is certainly
math already in what I've posted. Although a couplegroups 
have
dropped either iv or v of the series (at least from what
I'veseen on my server), there is already the beginnings of 
an
exposition onlattice theory. And my focus is on getting to
quantum logic, the study ofcausal sets, and logics for
reasoning about continuous dynamics here in thestart talking
about model satisfaction somewhere, to nish the
foundations,but I understand that physicist may still be wary
about applicability totheir own models, so I may invert the
discussion some.I intended only two threads so far. The rst
was to be towards aconstructive education, which was to be
quite largely cross posted (thoughwell within the scope of all
groups I posted to) in order to draw in theopinions from what I
knew to be a quite small minority in each eld,
butunfortunately found that newsservers have maximum
cross-posting limits, so Ineeded to repost in a chopped up
fashion, for which I appologise. Along theway, I came up with
the starting of an organised bibliography, which I triedto
post in the same thread as my rst. Unfortunately, I believe
one of thethe other two sections of posts for towards a
constructive educationgroups to post for this section, and
that particular posting software brokeit out of the thread
even though I replied within.Then I have this thread, the
anticlassicalist where I wanted to expand onthe things that
people had asked questions about in my rst thread.
Thissci.physics.research. It is meant to be educational as
well as provocative.It is certainly meant to raise the level
of discourse from the potty mouthedname calling so clogging
sci.physics, and is meant to introduce linguists tothe logical
formalisation which was so heavily questioned in my 
rstthread.
The logicians have had some arguments about these topics, so
Iwanted to enter the fray, and nearly all of my exposition
will bemathematical when I am not questioning psychological
motives.So, I believe my post targets are accurate, and I have
only managed to start4 threads total in the process. 2 of those
were incidental, not intended.Unlike cranks, I am open to
critical analysis and earnest debate.: > -=-=-=-=-= linguistic
negation =-=-=-=-=-=-: >: > The logic of natural language has
been studied by many different schoolsof: > thought throughout
history.:: Textual analysis: schools of thought do not study
anything, and is a: cliche.:: For somebody who is so
academically concerned with language, you seem: remarkably
insensitive to its use. Your earnestly learned turgidity: will
carry you far ... away from here.Ok, I change it to read only
schools. That was more my intention.As for turgidity, I do
understand that my writing can be up tight attimes, just as
sometimes it can be more playful. When I look to thesegroups,
however, I see that often the angry rebuttals cause either
crankishobstinacy or a learned timidity. The timidity response
places those wholaunched the angry rebuttals into authoritative
positions that often I donot feel they have the competency for.
Uncle Al is a prime example of onewho certainly is
knowledgeable, but is also quite ignorant and certainly notone
I see as an authority.I've studied obsessively for many 
years,
and will continue doing so for manymore. Any condence that
comes through in my posts is due to muchpractise, but I do not
want to convey that such condence excludes memaking mistakes
or accepting their being pointed out. I know I makemistakes
all the time. I focus on them, to try to understand them. But
Iwill not back down from my positions by content-less chest
beating.--
-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-galathaea:
prankster, fablist, magician, liar
===
Subject: Re: the
anticlassicalist }{ i: linguistic negation > Now really,
Ms.(?) fabulist, etc: there is much that is more grossly>
inappropriate to sci.physics here, so you might say all the
other kids> are doing it. But if you affect to be a _superior_
kid, might you not> lead by example and limit the number of
freshly cross-posted threads> which are mainly about
linguistics, and neither physics nor math?Here at sci.lang, we
sure haven't seen many posts about linguistics,other than 
one
indicating she'd once opened a volume of Proceedings.> You 
are
beginning to speak Crankish. > -=-=-=-=-= linguistic negation
=-=-=-=-=-=->> The logic of natural language has been
studied by many different schools of> thought throughout
history. Textual analysis: schools of thought do not study
anything, and is a> cliche. For somebody who is so
academically concerned with language, you seem> remarkably
insensitive to its use. Your earnestly learned turgidity> will
carry you far ... away from here. <snip>Evidence that she's
academically concerned with language?-- Peter T. Daniels
grammatim@att.net
===
Subject: Re: the anticlassicalist }{ i:
linguistic negatio
> They don't know nothing!>How bland.
Let's really mean it: They don't know nothing 
about nothing.>
The not-house will be built>Is this to mean the not-house
won't be not-built.Do even number of negations 
afrm and odd
number of negations deny? If he didn't not double negate,
didn't he afrm? If he didn't 
not double negate, didn't he not
afrm?How about double afrmatives implying 
negation?Aren't my
ideas the greatest?Oh yea, yea.
===
Subject: Re: the
anticlassicalist }{ i: linguistic negation: 2) are there also
well known schools of thought about negation fromAfrican: or
South American traditions?I didn't mean to appear to exclude
the su as a part of African thought,but intended other
schools of thought only because of familiarity.Syncretism,
particularly from gnostic schools, seems to be a fertile
groundfor such thought, but I was thinking more of the Ishango
bone or khipu andwondering of symbolic negation work in
societies in¤uenced by these andother early symbolic
mathematics.--
-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-galathaea:
prankster, fablist, magician, liar
===
Subject: Re: the
anticlassicalist }{ i: linguistic negation
>> I was looking
at litotes (perhaps more exactly, antenantiosis), with forms>
like>> He is not unattractive.> It is not impossible to do
that.>> where the double-negation implies some form of a shift
in meaning for> effect, following closely an empirical maxim
sometimes called the division> of pragmatic labor:>Do not
double negatives not leave you tied in knots?
===
Subject: Re:
the anticlassicalist }{ i: linguistic negation: Do not double
negatives not leave you tied in knots?It is eerie how I was
typing up a question about the khipu at the same timeyou
posted.Is it knot?(PS: Louis Kauffman has some beautiful stuff
on knot logics, as well.Reminds me of the BBC / TimeLife show
Connections with James Burke)--
-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-galathaea:
prankster, fablist, magician, liar
===
Subject: Re: the
anticlassicalist }{ i: linguistic negation > : Do not double
negatives not leave you tied in knots? It is eerie how I was
typing up a question about the khipu at the same time> you
posted. Is it knot?Anglice, <quipu>.> (PS: Louis Kauffman has
some beautiful stuff on knot logics, as well.> Reminds me of
the BBC / TimeLife show Connections with James Burke) -->
-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- galathaea:
prankster, fablist, magician, liar-- Peter T. Daniels
grammatim@att.net
===
Subject: Re: the anticlassicalist }{ i:
linguistic negation -=-=-=-=-= linguistic negation
=-=-=-=-=-=- The logic of natural language has been studied by
many different schools of> thought throughout history. Much of
this study has pointed to many models> different than those
based on classical logical structures. I have studied> this
eld for some time, but I have had some questions along the
way. I was looking at litotes (perhaps more exactly,
antenantiosis), with forms> like He is not unattractive.> It
is not impossible to do that. where the double-negation
implies some form of a shift in meaning for> effect, following
closely an empirical maxim sometimes called the division> of
pragmatic labor: The use of a longer, marked expression in
lieu of a shorter expression> involving less efort on the part
of the speaker tends to signal that the> speaker was not in a
position to employ the simpler version felicitously> (L. R.
Horn, Duplex negatio afrmat... the economy of double
negatives --> Chicago Linguistic Society 27-II)If you like
Larry's short paper, you might want to look at his book,
*ANatural History of Negation*. He's the leading expert on 
the
topic.-- Peter T. Daniels grammatim@att.net
===
Subject: Re: the
anticlassicalist }{ i: linguistic negation: If you like
Larry's short paper, you might want to look at his book, *A:
Natural History of Negation*. He's the leading expert on the
topic.The descriptions of that book look very, very promising,
suggestion; I will pickit up tomorrow!--
-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-galathaea:
===
prankster, fablist, magician, liarSubject: Re: the
anticlassicalist }{ i: linguistic negation
At least, Franz
Gnaedinger has something to say when he talks tohimself. Or
are you the latest avatar of Archimedes Plutonium?
===
Subject:
Re: the anticlassicalist }{ ii: the spectre
continues-=-=-=-=-= the spectre continues =-=-=-=-=-=-Now,
with this long history of logical analyses of language, I
alwaysfindit strange that there are comments like:Attention
being devoted here entirely to the classical two-valued
theoriesof truth-functions, quantiers, and identity, with
syntax, semantics, andpragmatics built upon their basis, there
will be no concern with alternativeforms of logic, so-called
three-valued (or, more generally, n-valued)logics, modal
logics, intuitionist logics, and the like. The view is
thatwhatever is valuable in these alternatives can be achieved
more readilywithin the classical framework by suitable
extensions.in the introduction to Semiotics and linguistic
structure by R. M. Martin,where such a representation is not
faithful to actual usage or expression.Now, I just recently
posed some questions on the newsgroups concerningnonclassical
logic, and certain linguists and physicists were actually
quiteconfrontational about the idea of educating about these
nonclassical logics.Expressibility was always proffered as a
reason, even though no one can evenclaim that boolean
is rich enough).So I still am left asking:3) what is the actual
source of the bias against nonclassical logics? Istill look at
classical logic and I see Aristotle and Western thought
inphilosophy and mathematics, but I don't see it 
representing
the way peopleverbally reason about the world usually. Why
must there be one and only onelogic of language? So much so
that there was an aggression in theresponses, as if not only
is that one useful, but that it is the one thatall people must
speak.Because I see many other, often non-Western, traditional
schools of meaningand language with much more faithful
representations of actual usage, and Iwonder why the bigotry?
As if the schools most in¤uenced by Aristotelianideas must
control the way we think about the world? I look to work
byKorzybski and see something ominous about this desire to
control theontology and mechanics of the way we reason about
the world. It isunderstandable where the Polish school of
logic and ontological theory comesfrom.And there is the
concordance between Aristotle's work in full, betweenlogic,
ethics, and an abstract monotheism, that has made his
workin¤uential in Jewish, Islamic, and Christian traditions,
but less so inpolytheistic traditions, so provocateur that I
am, I still believe there issomething here.The quote from
Martin even continues quite explicitly:Strictly, there is only
one logic, which, however, can be extendedvariously for 
specic
purposes as needed. .95One god, one country, 
onelogic,' is the
stunning phrase of Whitehead.The rst-order or elementary
point of view is pressed in this book, not asfar as possible
perhaps, but a good deal further than is usually
supposedpossible. The reader mayfind himself surprised to see
how much can beachieved on so narrow a basis. God made
rst-order logic and all the restis the handiwork of man.--
-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-galathaea:
===
prankster, fablist, magician, liarSubject: Re: the
anticlassicalist }{ ii: the spectre continues
galathaea
continues =-=-=-=-=-=- : : Now, with this long history of
logical analyses of language, I alwaysfind : it strangeWell,
it is not strange.You are ¤aunting ignorance here. : that
there are comments like: : : Attention being devoted here
entirely to the classical two-valued theories : of
truth-functions, quantiers, and identity, with syntax,
semantics, and : pragmatics built upon their basis, there will
be no concern with alternative : forms of logic, so-called
three-valued (or, more generally, n-valued) : logics, modal
logics, intuitionist logics, and the like. The view is that :
whatever is valuable in these alternatives can be achieved
more readily : within the classical framework by suitable
extensions. : : in the introduction to Semiotics and
linguistic structure by R. M. Martin, : where such a
representation is not faithful to actual usage or
expression.Actual usage or expression is just irrelevant.
Unless you are a linguistwhose object of study is how real
people use natural language, naturallanguage IS JUST STUPID,
because PEOPLE are just stupid. : Now, I just recently posed
some questions on the newsgroups concerning : nonclassical
logic, and certain linguists and physicists were actually
quite : confrontational about the idea of educating about
these nonclassical logics. : Expressibility was always
proffered as a reason, even though no one can even : claim
(lambda : calulus is rich enough).The parenthetical statement
is incredibly stupid. Lambda calculus is BIGGER AND BADDER
than classical boolean anything; it is MOREcomplex. SO OF
COURSE it is rich enough to expresswhatever. That is not even
the question. The questionis whether you can achieve similar
expressiveness withSIMPLER machinery (like classical boolean
algebra).There is, after all, a set of rst-order axioms
deninglambda-calculus.
===
Subject: Re: the anticlassicalist }{
ii: the spectre continues> There is, after all, a set of
rst-order axioms dening> lambda-calculus.Do 
you know of any
online presentations?:-)mitch
===
Subject: Re: the
anticlassicalist }{ iii: cognitive property maps-=-=-=-=-=
cognitive property maps =-=-=-=-=-=-However, I have also
learned that many are just unfamiliar with otherlogical
systems, so I wanted to also give a brief introduction to
lattices,logic, and models, to clarify my points.Lets start
with some objects we've collected. We have this ability
toabstract out properties of these objects that shows some
commonalities,Let's name them, by attaching a string of
symbols to each object andproperty we can identify (or pattern
recognise).Objects: # $ % $# $$Properties: $% %# %$ %%Now lets
say that we are able to form certain relationships between
theproperties and the objects, as in we can illustrate the
relationship of anobject having a certain property. We can do
this by drawing an arrow fromthe property list names to the
object list names that have the property.# $ % $# $$^ % % _*&
;%@| / / _ / { ~ /| / /_ / ~ ( /$% %# %$ %%Or, more clearly,
you have$% pointing to # and $%# ... % $# and $$%$ ... $# $$%%
$# $$Youfind a useful way to group these things by drawing 
more
arrows when allof an objects properties are also properties of
a second object, and in adual fashion whenever a property
points to all objects that another propertypoints to, because
this expresses the ability to tell the classicationtypes
apart. In other words,Both # and $ are pointed at by $% and
only $%, so they both point to eachother:# <-> $and maybe we
can't really think of the property that we thought told
themapart to name them differently. So we collapse them to one
name =.Since %# points to all three %, $#, $$ where %$ and %%
point to only $# and$$, we have $# <-> $$ ^ & % ^ | X | | / |%
$% <-> %% & ^ % | / | / %#and we realise we never were able to
tell $% and %% apart, just as we neverwere able to tell # and
$ apart, and similarly we admit that we cannot tell$# and $$
apart. You've forgotten what it was you had thought you saw
thatmade you name them different things, which happens when
you have so manythings to remember. The objects have all the
same properties ($% and %%),the properties all the same
objects.We collapse the objects and properties and no longer
make any distinctionbetween objects and properties; the two
collapsed names are chosen to be Aand E.Mathematicians would
say: A collapse starts with an equivalencerelationships and
takes what is known as a quotient by naming theequivalence
classes.Let's say that some smart, bunny-looking thing walks
up to you one day andconvinces you of the utility of drawing a
new arrow anytime you have twoarrows head to tail the same name
location. This bunny looking thing is amagician of great power
who calls this creation composition with eloquentarticulacy,
and you learn to see the bunny operation as giving the arrows
amotion or dynamics of meaning.What we are left with
is:=^||$%% E --> A & ^ % | / | / %#This seems to be a really
interesting way to think about the world, becauseit seems to
let us classify. We organise our thoughts and see
relationshipsbetween the various things we identify in the
world. Those arrows(admittedly poorly rendered in xed point
ASCII) or morphisms or whateveryou call them point from one
pattern you can recognise to another patternyou can recognise
that distinguishes itself by _adding_ recognition rules(ie.
the rules of the tail object are also rules of the head
object).And rules is just a name for properties or 
identiable
traits oranything that abstractly obeys the rules we establish
for it. There aremany ways of interpretation, and that freedom
allows us to read muchexpressiveness into these fundamental
relationships.Its a great way to map the world.We could even
start naming the arrows and having lots of arrows
betweenobjects where we suddenly get the richness of language,
of noun and verb, ofobject and transformation. However, lets
look more at what we have above.other name, there is always
either one or none arrows pointing from theformer to the
latter. And anytime we have no procedure to tell objectsapart,
we have collapsed them into a new name.Some hippy stoner dude
came along and convinced you that every name wasreally an
arrow pointing at itself, which he called the identity
morphism,maaaan. It doesn't make any difference whether you
think of objects ortheir identity morphism as primary since
they both can represent the samerelationships, so there's
nothing committed by the idea.What we've got, the
mathematicians call a skeletal pre-order catgeory or aposet.
We can call it whatever we want in our own pidgin, so I'll
justcall it a map with emphasis on the objects of cognitive
recognition. Theconnection between the acyclic digraphs found
in recognition problems andposets is detailed furthur by
Monique Pavel in Fundamentals of patternrecognition: 2nd
edition, revised and expanded.--
-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-galathaea:
prankster, fablist, magician, liar
===
Subject: Re: the
anticlassicalist }{ iv: from maps to logic-=-=-=-=-= from maps
to logic =-=-=-=-=-=-Now let us look at interpretting this
arrow symbol, such as in process,categorial inclusion, rule
renement, or progressions of state. We wantto be able to
evaluate logical relationships between objects. So we look
todene on a recognition map new objects dened 
by any two
objects whichallows us to collectively compare two objects
logically. The rst one ofthese, which can be called the 
least
upper bound or disjunction of twostates .95a' and 
.95b', written (a
/ b) which obeys the rules thata -> (a / b)b -> (a / b)forall x
in the lattice with a -> x and b -> x, we have (a / b) ->
xwhere it is important to differentiate when these are
derivationally foundon a lattice (objects already dened
through previous axioms with theappropriate universal property
derivable) and when the existence of newobjects is being 
dened
(for each new denition requires new consistency 
/completeness
/ etc. checks). Universals have interesting dualities
withexistential denitions (as seen in all of the 
there-exists
hidden abovewhen not derivationally found), but I don't want 
to
explore those here interms of the metalanguage. These
denitions are fairly safe, with manyconsistent models.These
disjunctions are unique after recognition collapse and capture
theidea that, whatever the arrow is interpreted as, there is an
object in thismap for any pair of objects with a universality
property of being pointedat by the pair and pointing at any
others also pointed at by the pair.It captures the conceptual
operations of joining concepts, collecting, orunion, and is
abstractly what is known as a coproduct.In linguistics, this
corresponds to an abstract representation of what iscalled in
English or (with inclusive semantics) when we are focused
onrecognition and property inclusion.There is also an
operation that has a duality to the coproduct by ¤ippingthe
arrows around and giving it the name .95/'. So the 
dention in
terms ofa pair of objects .95a' and 
.95b' is merely:(a / b) -> a(a
/ b) -> bforall x in the lattice with x -> a and x -> b, we
have x -> (a / b)with again existence implied when not
constructed.This operation has names like conjunction,
greatest lower bound, or moresimply product. It expresses a
description of where concepts meet intheir description. In
English, this is an abstraction of and in therecognition /
property inclusion interpretation.With the structure dening
product and coproduct objects for all pairs ofobjects in the
poset, we have what is known as a lattice. The lattice hasdual
connections or operations (/, /) with particular
universal(forall) relations in a variable over the lattice. An
algebra can beattached.These begin the logical theories of and
and or, conjunction anddisjunction, coproduct and product.--
-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-galathaea:
prankster, fablist, magician, liar
===
Subject: Re: the
anticlassicalist }{ v: universal truths-=-=-=-=-= universal
truths =-=-=-=-=-=-We can even dene another object _|_ with
the universal property:forall x in the lattice, we have _|_ ->
xand a dual object T withforall x in the lattice, we have x ->
T,where again it is important to point out the implied
existential axiom ifnot derivable. These give us two useful
identities on which to focus workin models. Yes / no
classication in models have been very 
benecial, andmuch of
our languages use them readily and often. But notice we still
havealot of freedom in the logic (there are no restrictions on
the existence ofother objects besides T and _T_, for example,
and many logics use otherobjects in this framework), and
models that lack these completeness objectsare still
useful.Let me give a little example of the calculus of
denition on these latticesthat can illustrate how the
lattices help us describe our ability to modelrelationships we
observe. Given a lattice, we can choose one object 
(let'scall
it P) standing for something in the ontology and add the axiom
thatT -> P. Then when we collapse the lattice to skeletal form
(since now T <->P and are the same thing represented under two
names, with .95T' kept), we maynd a 
cascade of collapses if P
pointed to any other objects (say Q1, Q2,...) wil each in turn
<-> T. So as we dene structure on the lattice, wecan 
represent
different models of the interactions of these universalobjects
(_|_ and T, whatever interpretation may be given them) with
otherpropositions on the lattice.And on these lattices we can
make more explicit our theories of negation.On sci.logic,
mitch recently gave a very interesting post on Stone
algebras.When we want to model uses of various denitions, we
need to show them asan axiom set. Stone algebras study the
axiom set:~(a) / ~(~(a)) <--> Twhere ~() is a mapping from the
lattice into itself. There are manyconsistent lattices that
have mappings that obey this axiom. As mitch'sinto ideals of
atomic boolean lattices which also classify throughembedding
pseudocomplemented distributive lattices, and that there is
aknown hierarchy between boolean lattices and
pseudocomplementeddistibutive lattices which include the
lattices that obey the Stone axiom.So this axiom denes a
class of logics somewhere between the boolean worldand
lattices that are pseudocomplemented and distributive.These
two terms are dened as structural axioms of the lattice. A
latticethat is pseudocomplemented has dened for every object
.95a' another 
object'a*' with the propertyforall x in the
lattice with (x / a) <-> _|_, we have x -> a*whereas a lattice
that is distributive obeys the axiomfor all x, y, and z in the
lattice, we have x / (y / z) <-> (x / y) / (x/ z)and we can
start to attach meanings to these kinds of lattices
byunderstanding the universal properties implied. Theyfind
uses in trigeringcollapse operations on the lattice during
derivations or deductions in thestructure. The
pseudocompliment gives a universal disjoint property
tonegation. Distributivity gives a well known algebraic
relationship betweenany three objects of the lattice, and I
will later show that quantum logicsare examples of logics that
are not distributive.So we start to see the rich world of
lattices and meaning, slowly. Othernegations than those
mentioned above can be explored. There are
subminimal,preminimal, and minimal negations. There is de
Morgan, intuitionistic, andortho negation.Still so much to
describe...--
-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-galathaea:
prankster, fablist, magician, liar
===
Subject: HELL :-) (Was:
Re: the anticlassicalist }{ ii: the spectre continues)> On
<galathaea@excite.com> The quote from Martin even
continues quite explicitly:>> Strictly, there is only one
logic, which, however, can be extended> variously for
specic purposes as needed. .95One god, one 
country, one>
logic,' is the stunning phrase of Whitehead.>> Nonsense.
This is just a description of HELL! :-)lolYou never did
understand why I was posting to sci.logic. :-)In Process and
Reality Whitehead had been kind enough to say that
mathematicsset a bad example for philosophy. Of course, his
discussion of extensiveconnection in later chapters relegated
that to a vacuous nonsequitur. But, onemight say that he tried
to not...????When professionals in every academic discipline
involve their topic withmathematics as a justicationalist
tool, people with a sense of the subject mustinvariably
suffer. Imagine trying to make sense of inconsistent
mathematics,
http://plato.stanford.edu/entries/mathematics-inconsistent/
However, specifying a topological space by itsclosed sets is
every bit as reasonable as specifying itby its open sets. Yet
the logic of closed sets is knownto be paraconsistent, ie.
supports inconsistent theories;Naturally while the academic
jackasses argue about these matters at faculty tea,they
continue to teach nonsense (=inconsistent) to their students.
Caveatemptor. It is all too easy when one has refuge in the
pathetic retort, But, *I*am not talking about THAT!As an
example of the dark side with this issue, let me consider
Grothendieck for amoment. Pierre Cartier has a lovely paper
about space and symmetry
http://modular.fas.harvard.edu/sga/from_grothendieck.pdn
months inprison and a ne of 20,000 francs. It seems to 
methat
his denitive break with science dates from thisincident. He
withdrew more and more into his owntent. After 1993 he no
longer had a postal addressand settled into a hamlet in the
Pyrenees. Very fewmanaged to see him there. If I can believe
his mostrecent visitors, he is obsessed with the Devil, whomhe
sees at work everywhere in the world, destroyingthe divine
harmony and replacing 300,000 km/sec by299,887 km/sec as the
speed of light!It is terribly easy to relegate this to
nonsense--but, after all, we are talkingabout HELL here.
:-)There is an embedding theorem in combinatorial topology
stating that everyn-dimensional triangulation is isomorphic to
a triangulation in 2n+1 dimensions.Currently, there appears to
be a group of theoretical physicists advocating atheory
embedded in the symmetries of a 26-dimensional Lorentzian
lattice. So,here is a little calculation for Grothendieck's
Devil,300,000299,887------- 113Using the number 26, we
have(26)...(1)...(26) 53 2n+1 2 2-monotonic ordering constants
--- 55(55)...(1)...(55) 111 2n+1 2 2-monotonic ordering
constants --- 113And, of course, I just made a different post
mentioning the logic of apseudocomplemented lattice in which
everything becomes true after no more than twoapplications of
pseudocomplementation,-----...and as a pseudocomplemented
distributive lattice with pseudocomplementa* | 0 if a <> 0
Chains with 0, 1 : a* = | | 1 if a = 0-----Unlike Cartier, I
recognize a certain rationality behind the reported fact.
Thatis not to say that I concur with the conclusion,
however.One wants to immediately say that mitch is engaging in
nonsense, trickery, ornumerology. Well, perhaps we should
overlook the fact that* INCONSISTENCY SHOULD BE
NONSENSE**MATHEMATICS IS FULL OF TRICKS**REDUCING TOPICS TO
THE SCIENCE OF NUMBER IS NUMEROLOGY*Regardless, if we are
going to accept mathematical principles in any case, weshould
not be ignoring the statistical entropy described by Shannon
and the factthat there is almost no understanding of
analog-to-digital quantization inmultiple dimensions (per
Sphere Packings, Lattices and Groups by Conway andSloane). In
my opinion, that is what seems to be involved here.Now, I will
gladly admit to being confused. There is plenty of evidence of
thaton sci.logic in the past year.And, I will gladly admit to
talking in circles. Indeed, I have been accused ofthat often
enough as well.But, let me share a non-circular observation
with everyone. Whereas I thought theappearance of 26
dimensions might have had to do strictly with the ubiquity of
theEnglish alphabet in technical literature, I was completely
astonished when Irecently opened my Bible on a lark. Turning
to the Book of Numbers, the prefacereads---The Book of Numbers
derives its name from the account ofthe two censuses of the
Hebrew people taken one near thebeginning and the other toward
the end of the journey in thedesert (chapters 1 and 26)---Once
again, this is not about numerology. It is about the structure
onformation. At
http://www.research.att.com/~njas/lattices/ANABASIC_11.htmlyou
willfind the description of an 11-dimensional lattice for 
which
no subset ofminimal vectors forms a basis. The Gram matrix for
this lattice is 60 5 5 5 5 5 -12 -12 -12 -12 -7 5 60 5 5 5 5
-12 -12 -12 -12 -7 5 5 60 5 5 5 -12 -12 -12 -12 -7 5 5 5 60 5
5 -12 -12 -12 -12 -7 5 5 5 5 60 5 -12 -12 -12 -12 -7 5 5 5 5 5
60 -12 -12 -12 -12 -7-12 -12 -12 -12 -12 -12 60 -1 -1 -1 -13-12
-12 -12 -12 -12 -12 -1 60 -1 -1 -13-12 -12 -12 -12 -12 -12 -1
-1 60 -1 -13-12 -12 -12 -12 -12 -12 -1 -1 -1 60 -13 -7 -7 -7
-7 -7 -7 -13 -13 -13 -13 96As can plainly be seen, the number
60 (that is, the base of the sexigesimal numbersystem used by
ancient Sumer) lies on the diagonal in every position except
thelast. As for the number 96, it plays an essential role in
quantum logic. On page5 Jacek Malinowski's paper,
http://www.uni.torun.pl/~jacekm/swc1.pdfyou will see mention
of the Chinese lantern. This is a six element
orthomodularlattice. The description of a quantum consequence
operation depends on the crossproduct of this structure with
the sixteen standard Boolean switching functions.This is
discussed in Protoalgebraic Logics by Janusz Czelakowski,
although theproof is omitted.Naturally, this has very little
to do with interpreting physics--there are nooperators on a
Hilbert space such that PQ-QP=1. But, it might indicate the
properinterpretation of quantum logics with respect to
information theory--if there wasanyone paying close enough
attention.But, like Pavlov's dogs, everyone has been trained
to say, *I* don't donumerology.Oops... too late.For those
interested in that kind of thing, one might compare the Gram
matrixabove with the story about separating the rocks when
crossing the Jordan riverinto the Promised Land. But, keep in
mind that the mathematical comparison shouldbe the fact that
convex deltahedra can have a maximum of 20 sides. A
60-sideddeltahedron is not convex. The story is about
discarding information. One cannd the relationship between
the sexigesimal number system and the Great PlatonicYear in
the work of Joseph Campbell.Anyway, if I apply the same sort
of construction as with Grothendieck'srecalculation, I might
write,(11)...(1)...(11) 23 2n+1 2 2-monotonic ordering
constants --- 25And, coincidentally, Ifind a 25 member 
listing
in Conway and Sloane. What isparticularly interesting is that
the endpoints are xed at 0 and 1, as is neededby classical
logic. n lattice density 0 0 1 1/2 2 1/(2*surd(3)) 3
1/(4*surd(2)) 4 1/8 5 1/(8*surd(2)) 6 1/(8*surd(3)) 7 1/16 8
1/16 9 1/(16*surd(2))10 1/(16*surd(3))11 1/(18*surd(3))12
1/2713 1/(18*surd(3))14 1/(16*surd(3))15 1/(16*surd(2))16
1/1617 1/1618 1/(8*surd(3))19 1/(8*surd(2))20 1/821
1/(4*surd(2))22 1/(2*surd(3))23 1/224 1Anyone with the
slightest grasp of how Lie algebras are applied by
generalizationto answer questions in quantum physics should
look at the triples for n=(9,10,11)and n=(13,14,15). The graph
K_(3,3) is non-orientable, and, the logical analysisof quantum
operators by von Neumann and Birkhoff led to Mackey's 
seventh
axiom.That is, the logic of quantum mechanics suggests a
framework of innitely manyquestions. I can see it now...Give
me 3.Give me 3 more.Give me 3 more.Give me 3 more.ad
innitum.I know that it is difcult to 
understand my
point--after all, people talking incircles leave little room
for a rational person to get a word in edgewise.Almost
everyone *knows* that a sufcient amount of text will permit
somenegative (apocalyptic) conclusion under statistical
Positivity,Total positivity is a concept of considerable power
thatplays an important role in various domains of
mathematics,statistics, and mechanics. In mathematics, totally
postivefunctions gure prominently (though sometimes
indirectly)in problems involving convexity, moment spaces,
eigenvaluesof integral operators, and the oscillation
properties o¤inear differential equations, as well as in the
theory ofapproximations and other areas of real analysis. In
statisticsthe theory of total positivity is fundamental to an
understandingof statistical decision procedures, and
especially in discerninguniformly most powerful tests for
hypotheses involving anite set of real parameters. Total
positivity is also ofgreat importance in ascertaining optimal
policy for inventoryand production processes, in evaluating
the reliability ofcoherent systems, in the analysis of
diffusion-type stochasticprocesses, and in the study of
coupled mechanical systems.one begins to realize that
mathematical pictures are easily biased. This seems toresult
from picture-thinking and can be expressed formally in terms
of theisoperimetric inequality,---The area F and the length L
of any plane domain with rectiableboundary satisfy the
inequality L^2 - 4(pi)F >=0;the equality sign holds only in
the case of a circle---At least the physicists seem to be
concerned with one particular circle. Theequation for the ne
structure constant, Alpha, is of the form, e^2 4(pi)Alpha =
--------------------- (epsilon_0)(h-bar)(c)So, unlike
Grothendieck, I have no particular concern about the speed of
lightchanging. But, circles are convex domains and it is the
mathematics of convexitythat is ubiquitous.The more I
investigate the history of this subject, the more clear it is
thatthese issues, mathematics has the soma effect a feel-good
generation desires.But, someone has to have made a mistake
unless there is no such thing as reality.My experience with
closed physical systems is that there are conservation
laws.:-)mitch
===
Subject: JSH: Question of intentI've been
thinking about formalizing the argument with which I prove
aproblem with the ring of algebraic integers, and ran into a
problem.I'd envisioned a proof along the lines of, given
algebraic integerfunctions f_1(x), and f_2(x), where
f_1(0)=f_2(0) = 0, and algebraicinteger function P(x), also a
primitive polynomial, where(f_1(x) + 7)(f_2(x) + 1) = 7P(x)it
must be true that f_2(x) + 1 is coprime in general to 7
implyingthat f_1(x) has 7 as a factor in general, but as it
does not in thering of algebraic integers, there is a problem
with the traditionalunderstanding of that ring.The only
problem is that thinking of it that way I realized that
it'snot true, since you *can* have algebraic integer 
functions
that neverequal 0, which are themselves factors of 7.For
instance, you can just take the roots ofy^2 + (b+8)y + 7.Now I
posted somewhat on this subject earlier as the issue here
reallyis the ability of someone to come along and use certain
functions tomultiply times any factorization and get something
interesting.However, that requires intent, since there are an
innity of suchfunctions available, even to factor 7 into two
ways, so how do youchoose?A human being can pick functions for
any number of reasons, and itwouldn't surprise if they 
picked
something easy, like the roots ofy^2 + (b+8)y + 7.So what's
different with my argument? The answer is that multiplyingby a
function means you have multiplied by that function, and
hidingthat fact requires more intent.Like in my earlier post,
if I have w_1(x)(f_1(x) + 1), then you havew_1(x) f_1(x) +
w_1(x)and to hide that fact and appear to have a free contant
term requireswrapping things up quite deliberately,
like((w_1(x)(f_1(x) + 1) - 1) + 1and using say, a_1(x) =
(w_1(x)(f_1(x) + 1) - 1)to geta_1(x) + 1so that it *looks*
like there's a free constant term of 1.Now 
it's easy to
demonstrate that, but regardless of how it was made,a_1(x)
*is* still an algebraic integer function!The issue here then
is that you can have functions without trueconstant terms,
while I rely onfinding constant terms.However, it takes
intent, as there are an innity of such examples,and if a
human being isn't trying to hide functions by wrapping 
themup
like I showed above, then you can simply separate off any
suchfactor like w_1(x).I'm still thinking about how to go
about saying that formally:Given algebraic integer functions
f_1(x), and f_2(x), wheref_1(0)=f_2(0) = 0, and algebraic
integer function P(x), also aprimitive polynomial,
where(f_1(x) + 7)(f_2(x) + 1) = 7P(x)where 7 is a truly a
constant term of f_1(x) + 7, and not a made upone for effect?
How do you handle human intent mathematically?James
Harris
===
Subject: Re: JSH: Question of intent> How do you
handle human intent mathematically?Easy.IF .95mathematics is
posted by JSH'THEN .95intent is to aggrandize self 
and afrm own
magnicence'ELSE .95intent is 
irrelevant'> James The Magnicent,
W.O.O.O Harris--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: JSH: Question
of intentX-DMCA-Notications:
http://www.giganews.com/info/dmca.html>I've been thinking
about formalizing the argument with which I prove a>problem
with the ring of algebraic integers, and ran into a
problem.Guffaw! Of course you did. It's probably similar to
the problemsI run into every time I try to formalize my proof
that the Earthis ¤at.>[...] How do you handle human intent
mathematically?You don't - mathematical proofs do not 
involve
any such notion.>James Harris************************David C.
Ullrich
===
Subject: Re: Question of intent (PG13)James Harris>
I've been thinking about formalizing the argument with which 
I
prove a> problem with the ring of algebraic integers, and ran
into a problem....Meaning that another absurd claim of yours
was trivially refuted.I know lots of self-interested liars,
Harris, but I fail to see who it isyou hope to deceive with
this  of yours -- page after page about ain' 
quadratic
polynomial. Don't you get tired of making such an ass
ofyourself? You never succeed in deceiving anyone, and you
prove nothingexcept your own ignorance, over and
over.
===
Subject: Re: Difference between polynomial and eld
arithmetic!!Thanx for your explanation.. but I really don't
understand thedifference in calling them Polynomial arithemtic
and eldarithemtic.. as, for example, when i do a polynomial
addition whosecoefcients are eld elements..say 
in GF(2^m) I
am basically addingthe eld elements ..(adding 
coefcients for
that particularpower)thus making it eld arithemtic.. similar
case formultiplication....so what I don't understand is how
come it ispolynomial arithmetic and is more complex.. compared
to eldarithmetic???i hope i clearly explained my question. 
any
difference between eld>arithmetic and polynomial
arithmetic.>>Any help would be greatly appreciated.>>
make more sense to ask> whether there is any connection
between the two, rather than what is> the difference between
them. Many elds are of the form F(a), where F is some more
basic eld.> Typically, F is either a nite 
eld F_p of
integers mod p, or F is> the eld of rational numbers, and 
you
assume that you can already do> arithmetic in F. The element a
could be either transcendental or algebraic over F.> In the
rst case, F(a) is isomorphic to the eld of 
rational
functions> over F, so arithmetic in F(a) can be carried out
using polynomial> arithmetic. If a is algebraic over F, then
it satises a minimal polynomial f(a) = 0,> for some
polynomial f, and then arithmetic in F(a) can be carried out>
using polynomial arithmetic modulo f. But for reasonably small
nite elds, there are other methods of doing> 
arithmetic,
which do not involve polynomial arithmetic, which can be more>
efcient. One way is to choose a primitive element b of the
eld, and> represent nonzero elements x by the integer n,
where x = b^n.> Then multiplication is very fast, and addition
can be carried out using> a stored pre-computed table of Zech
logarithms l(n), dened by> b^n + 1 = b^{l(n)}. Derek
Holt.
===
Subject: Re: PDE Max-Min Principle (Help?)Mohan Pawar
question really about math...>> Prove that if u(x,t) <=
v(x,t) for x=0, for x=l (lower-case L), for>> t=0, then u<=v
for all (x,t) in 0<=x<=l, 0<=t<=oo. Given> u(0,0) <= v(0,0)>
u(L,0) <= v(L,0)I think it is saying:Givenu(0,t) <=
v(0,t)u(L,t) <= v(L,t)u(x,0) <= v(x,0)> Prove that> For
certian class of PDEs , u(x,t) <= v(x,t) for all (x,t) in>
0<=x<=L, 0<=t<=oo Is this correct interpretation of your
question? I am missing> something. Mohan Pawar> - Tim--
Timothy M. BrauchGraduate StudentDepartment of MathematicsWake
Forest University
===
Subject: Re: PDE Max-Min Principle
(Help?)The World Wide Wade
u(x,t) <= v(x,t) for x=0, for x=l (lower-case L), for>> t=0,
then u<=v for all (x,t) in 0<=x<=l, 0<=t<=oo. Ridiculous as it
stands. Please include all hypotheses. (Also, what> does t = oo
mean here?)> Sorry, at the beginning of the problem set, we
were told that u and v are are solutions to the diffusion
(heat) equation u_t = k*u_xx.There is a hint that says this is
proving the Comparison Principle using the Maximum-Minumum
Principle.Also, that last inequality should read 0<=t<oo, it
doesn't equal innity.Like I said, 
I'm not looking for the
answer, just a way to start the beast. Most other problems in
this set were proving properties of the diffusion equation
using change of variables and the chain rule. That does not
seem to work here. - Tim-- Timothy M. BrauchGraduate
StudentDepartment of MathematicsWake Forest
University
EqlO6oZlQeXHFQy1W16zEhRl+es401N2bojDIZu7g0-RmG9Z4LZtYGTim
that if u(x,t) <= v(x,t) for x=0, for x=l (lower-case L), for>
>> t=0, then u<=v for all (x,t) in 0<=x<=l, 0<=t<=oo.>>
Ridiculous as it stands. Please include all hypotheses. (Also,
what> does t = oo mean here?)> Sorry, at the beginning of
the problem set, we were told that u and v> are are solutions
to the diffusion (heat) equation u_t = k*u_xx. There is a hint
that says this is proving the Comparison Principle> using the
Maximum-Minumum Principle.can you apply the Maximum-Minumum
Principle. to w=u-v ? which PDE and boundaryconditions does w
fulll?hthklaus Also, that last inequality should read
0<=t<oo, it doesn't equal> innity. Like I 
said, I'm not
looking for the answer, just a way to start the> beast. Most
other problems in this set were proving properties of the>
diffusion equation using change of variables and the chain
rule. That> does not seem to work here. - Tim --> Timothy M.
Brauch> Graduate Student> Department of Mathematics> Wake
Forest University
===
Subject: Re: PDE Max-Min Principle
(Help?)The original post has some data missing and I am hoping
that the hint belowhelps.Assume PDE u_t = u_xx for t >0, 0 < x
< LNote that the given IC impliesIC => u(x,0) - v(x,0) = w(x)
for 0 =< x =< LNow the given BCs also implyBC1 => u(0,t) -
v(0,t) = k1, k1 > 0, t >0BC2 => u(L,t) - v(L,t) = k2, k2 > 0,
t >0Hint:Begin withu(x,t) - v(x,t) = w(x) => v(x,t) = u(x,t) -
w(x)Convert given original IBV problem in u into a new IBV
problem in v. Anintermediate simple BVP is created in the
process.If I didn't mess up the simple BVP should look
likew(x)=0w(0)=k1w(L)=k2solve this for w(x). Tie loose ends
and make the argument leading tou(x,t) - v(x,t) > w(x) for t
>= 0, 0 =< x =< L------------------Mohan
Pawar------------------klaus hoffmann
that if u(x,t) <= v(x,t) for x=0, for x=l (lower-case L), for>
>> t=0, then u<=v for all (x,t) in 0<=x<=l, 0<=t<=oo.>>
> Ridiculous as it stands. Please include all hypotheses.
(Also, what> does t = oo mean here?)>>> Sorry, at
the beginning of the problem set, we were told that u and v>
are are solutions to the diffusion (heat) equation u_t =
k*u_xx.>> There is a hint that says this is proving the
Comparison Principle> using the Maximum-Minumum Principle.>
can you apply the Maximum-Minumum Principle. to w=u-v ? which
PDE andboundary> conditions does w fulll?>> hth> klaus>>>
Also, that last inequality should read 0<=t<oo, it doesn't
equal> innity.>> Like I said, I'm not looking 
for the
answer, just a way to start the> beast. Most other problems
in this set were proving properties of the> diffusion
equation using change of variables and the chain rule. That>
does not seem to work here.>> - Tim>> --> Timothy M.
Brauch> Graduate Student> Department of Mathematics>
Wake Forest University
===
Subject: Re: How tofind near integer
values of n*a and n*b?israel@math.ubc.ca (Robert B. Israel)
efcient way to solve the following problem:|> Given 
a,b,eps>0
nd all positive n<eps^-2 such that n*a and n*b are>within
eps>|> of being integer.|> For example, let a=sqrt(2),
b=sqrt(3) and eps=10^-k. I've compiled a>table of>|> n that
work for various small values of k. Is it complete? >>Given
any m real numbers x_1,...,x_m, there exist innitely many
positive>integers n such that n x_j are all within n^(-1/m) of
integers. But I don't>think there's a lot more 
known on this
topic. > I'm curious if there might be some *reason* to 
think
the following is true:Consider the set X=X(x_1,x_2)={n:n*x_1
and n*x_2 are within n^(-1/2) ontegers}. Is there a constant
k such that for each n in X there exist n_i inX and integers
a_i>0 with n=sum(a_i*n_i) and sum(a_i)<=k?Unless I have made
an error, if x_1=sqrt(2) and x_2=sqrt(3) and n<=1580818405then
k=10 always works and k=2 usually works. Some examples of this
(usingRI's data) are the following:>> d, n, eps> 5, 41,
0.017243942841=22+19> 6, 1463, 0.00966852671463=1422+41> 7,
4109, 0.00352779114109=2646+1463> 8, 20586,
0.002075387020586=16477+4109> 9, 326491,
0.0002139702326491=171484+155007> 11, 2151016,
0.00010642382151016=1824525+326491> 13, 78411940,
0.00004235278411940=51448791+26963149> 14, 447810523,
0.0000111850447810523=237386836+210423687> 15, 1580818405,
0.00000546921580818405=1133007882+447810523> 16, 10011132893,
0.000001018710011132893=8430314488+1580818405So it seems as
though many elements of X(sqrt(2),sqrt(3)) can be written as
asimple sum of two other members of X (i.e. k=2, often).
However, if n=134421 then n=6*20586+2*4109+2646+41, so k>=10.
Is there anyreason (otwt) to believe that k=10 might work for
all n in X?Rich
===
Subject: Re: Bound of a sum>>I admit I am
no math genius, but I need tofind some way to bound>>k-1 n>
>>sum sum 2/(j-i+1)>>i=1 j=k+1>>Any upper bound that is
some constant by n is OK.>Write the sum as >S =
sum_{r=3}^n f(r,k)/r >where f(r,k) = min(k-1,n-r+1) -
max(0,k-r+1).> Oops, somehow I lost the 2. So my bound would
be 2n.> Robert Israel ==Let k-1 n S(n,k):= sum sum 2/(j-i+1) ,
where 1< k < n . i=1 j=k+1In the following a,b,r,s,n denote
positive integers, and h(n)=1+1/2 +1/3+ ....+1/n , g_r(a)
=h(r*a) - h(r)J.Lambek and L.Moser, see ,,Rational analogues
of the logarithm function,Math.Gaz. 40 (1956), 5-7, have
established the inequalities a/(a+b) =< h(a+b) - h(b) < a/b ,
0=< g_{r+1}(a) -g_r(a) < 1/r -1/(r+1) ,(1) 0=< g_s(a)-g_r(a) <
1/r -1/s if r<s , 0=< g_r(a*b)-g_r(a) -g_r(b) < 1/r , 1/(a+1)
=< g_r(a+1) -g_r(a) < 1/a .Also , following inequalities hold
: 1/n +ln(n) < h(n) < 1+ln(n)(2) gamma < h(a)+h(b)-h(a*b) =< 1
, [Amer.Math.Monthly 55 (1948),317, Problem E819],,,gamma being
Euler's constant.I found equality(#) S(n,k)= 2*SUM_{i=1 to
i=k-1}( h(n-k+i+1) - h(i+1) ) .Using Abel's identity 
SUM_{i=1
to i=k-1}A(i)*B(i)= = A(k-1)*C(k-1) + SUM_{i=1 to
i=k-2}(A(i)-A(i+1))*C(i)where, C(i)=SUM_{j=1 to j=i}B(j) ,
from (#) onefinds (#')
S(n,k)=2*(3+(n+1)*h(n)-(n-k+2)*h(n-k+2)-(k+1)*h(k) ) .Using
(1)-(2) in (#') it's possible to 
nd bounds for S(n,k).==I
appreciate that another method to bound S(n,k) is to observe
that 1/j =INTEGRAL_{t=0 to t=infty} e^{-j*t} dt , j=1,2,...
,=or to use Psi-function, Psi(x):=Gamma'(x)/Gamma(x) 
.Perhaps
help, Alex
supports the slightly stronger>> statement that e+pi and e*pi
remembered>correctly - or they lack someting :-)Let u=e+pi, and
v=e*pi.Also, u and v are both positive, and e < pi.Let uu =
u/2.v = (uu+x)*(uu-x) = uu^2 - x^2.Hence, x = v - uu^2.And e =
uu-x, pi = uu+x.They *had better* not be both algebraic, or e
and pi would be too!John
Savardhttp://home.ecn.ab.ca/~jsavard/index.html
===
Subject: Re:
21:37:11 -0500, Michael N.
is the origin of space and time?>> Space and time is the
4D fabric within which all life>> has evolved.>> Wrong. It
is in fact the 8th dimentional time-space fabric.>> See .95The
adventurures of Buckaroo Banzai across the 8th dimension'.>>
Do your homework, kid!>>Uh, I think it is you who needs to do
the homeowork. Everyone knows there>are exactly FIVE
dimensions. Every hear about a little something
called>'Twilight Zone' (aka the 
fth dimension) !?!?! I bet
you feel pretty dumb>right about now.Actually, we're not 
sure.
There may be 10 or 11 dimensions, dependingon whether you go
with superstrings or supergravity. But there's a newtheory
that says the extra dimensions aren't curled up into
littleballs or toruses after all, and the Universe is held in
four of thosedimensions through a different mechanism...On the
other hand, I remember once from an old Flash comic book
wherethe Flash and Kid Flash visited a world with 24
dimensions, inhabitedby tall grey beings who looked very
three-dimensionally humanoid...John
Savardhttp://home.ecn.ab.ca/~jsavard/index.html
===
Subject: Re:
is the origin of space and time?Where and when could space and
time have originated, if there was nospace and time before
they originated?John
Savardhttp://home.ecn.ab.ca/~jsavard/index.html
===
Subject: Re:
+0100,
the role of innity in math:The role of completed 
innity is
limited to stuff like axiomatic settheory, but incomplete
innity pervades mathematics.>How is it 
dened?>>Why is it
needed?>>At what precition does math work?If math were limited
to any set precision, it would be merecomputation, and if a
higher precision were needed, then we would haveto work out
math all over again.That's why innity is 
needed in math, so
we get it right the rsttime.John
Savardhttp://home.ecn.ab.ca/~jsavard/index.html
===
Subject: Re:
The role of innity in mathjsavard@ecn.aSBLOKb.caNADA.invalid
limited to stuff like axiomatic set> theory, but incomplete
innity pervades mathematics. What's incomplete 
about the
innity of the real numbers?
===
Subject: Re: Mathcad Upgrade
QuestionMathcad 11 is worth the upgrade but I have found that
it doesn't workreliably on WIN98. I upgraded to WINNT and it
works ne. I also hatec-dilla, but it should do anything bad
to your system unless you have somesoftware that scans for
trojans and stealth programs. I would rather havea USB
dongle.Peter NachtweyRobert Wynne <robertwynne2@comcast.net>
2001i
this weekend. After> installition I noticed that c-dilla had
also been introduced to myregistry.> Google c-dilla tofind 
out
what this program does. Anyhow I have spent 2> days rebuilding
my hard drive because of the damage that Mathcad did.> Steven
in> I am thinking of upgrading from an old version of Mathcad
(Version 7)> to the most recent. But past experience with
upgrades has taught me> to be a bit wary, so 2 questions:>
>> 1. Did Mathsoft do anything to screw up the product, in
terms of> usability or features, since I last purchased it
(back in 1997)?>> 2. Does the latest product come with any
kind of registration> feature, like Mathematica has, that
locks your software into one> computer, and possibly locks
you out of the software if you upgrade> your hardware? (I
respect the rights of companies to make money off> their
software, but I'm a rm believer that the only 
fair deal is>
one-user/one-app. If I have three computers -- and I do -- I
should> not have any worries or complications with loading
the same software> package on all three of them.)>>
Standard Anti¤ame Disclaimer: Please don't ¤ame me. I 
may
actually> *be* an idiot, but even idiots have
feelings.>>
===
Subject: Re: Mathcad Upgrade QuestionYes spybot
caught on to it. Besides I don't like folks putting stuff on
mycomputer without telling me rst.Neither Mathematica nor
Matlab has any problems with the way I use mycomputer. Does
Mathcad 11 usec-dilla to keep track of what your doing? Or did
the upgrade but I have found that it doesn't work> reliably 
on
WIN98. I upgraded to WINNT and it works ne. I also hate>
c-dilla, but it should do anything bad to your system unless
you have some> software that scans for trojans and stealth
programs. I would ratherhave> a USB dongle.>> Peter Nachtwey>>
Don't use mathcad. I installed version 2001i this weekend.
After> installition I noticed that c-dilla had also been
introduced to my> registry.> Google c-dilla tofind out what
this program does. Anyhow I have spent 2> days rebuilding my
hard drive because of the damage that Mathcad did.> Steven O.
> I am thinking of upgrading from an old version of Mathcad
(Version 7)> to the most recent. But past experience with
upgrades has taught me> to be a bit wary, so 2 questions:>
>> 1. Did Mathsoft do anything to screw up the product,
in terms of> usability or features, since I last purchased
it (back in 1997)?>> 2. Does the latest product come
with any kind of registration> feature, like Mathematica
has, that locks your software into one> computer, and
possibly locks you out of the software if you upgrade>
your hardware? (I respect the rights of companies to make
money off> their software, but I'm a rm 
believer that the
only fair deal is> one-user/one-app. If I have three
computers -- and I do -- I should> not have any worries or
complications with loading the same software> package on
replies.>> Steve O.>> Standard Anti¤ame
Disclaimer: Please don't ¤ame me. I may actually> *be* 
an
idiot, but even idiots have feelings.>>
===
Subject: Re:
Mathcad Upgrade QuestionYou may be coming up against the old
saying If it isn't broken, don't 
xit!. I have MathCad 2000
professional, which works very well and haveresisted the UK
seller's efforts to get me to do several upgrades. It 
soundsas
though that was the right decision.I also have Delphi 3, 5 and
7, Personal, on my PC. They all work and 5 wasslightly better
than 3 but 7 seems less good than 5. I suspect a lot
ofupgrades are more trouble than they are worth.Robert Wynne
on to it. Besides I don't like folks putting stuff onmy>
computer without telling me rst.> Neither Mathematica nor
Matlab has any problems with the way I use my> computer. Does
Mathcad 11 use> c-dilla to keep track of what your doing? Or
worth the upgrade but I have found that it doesn't work>
reliably on WIN98. I upgraded to WINNT and it works ne. I
also hate> c-dilla, but it should do anything bad to your
system unless you havesome> software that scans for trojans
and stealth programs. I would rather> have> a USB dongle.>
>> Peter Nachtwey>> Robert Wynne
mathcad. I installed version 2001i this weekend. After>
installition I noticed that c-dilla had also been introduced
to my> registry.> Google c-dilla tofind out what this
program does. Anyhow I have spent2> days rebuilding my
hard drive because of the damage that Mathcad did.> Steven
in> I am thinking of upgrading from an old version of
Mathcad (Version7)> to the most recent. But past
experience with upgrades has taught me> to be a bit
wary, so 2 questions:>> 1. Did Mathsoft do
anything to screw up the product, in terms of> usability
or features, since I last purchased it (back in 1997)?>>
> 2. Does the latest product come with any kind of
registration> feature, like Mathematica has, that locks
your software into one> computer, and possibly locks you
out of the software if you upgrade> your hardware? (I
respect the rights of companies to make money off> their
software, but I'm a rm believer that the only 
fair deal is>
> one-user/one-app. If I have three computers -- and I do --
I should> not have any worries or complications with
loading the same software> package on all three of
>> Steve O.>> Standard Anti¤ame Disclaimer:
Please don't ¤ame me. I mayactually> *be* an idiot, but
even idiots have feelings.>>>>
===
Subject: 1 to 1
correspondenceHow would I show a 1 to 1 correspondence between
(0,1) and [0,1] ?????
===
Subject: Re: 1 to 1 correspondence>
How would I show a 1 to 1 correspondence between (0,1) and
[0,1] ?????if you just need to show that (0,1) and [0,1] have
the samecardinality (i.e. just the existence of the 1 to 1
correspondance) youmay do the following:1- consider the
half-circle, of radius 1 and center P, tangent to thereal
line. This half circles represents the interval (0,1) (0 on
oneend, 1 on the other). draw a line starting from P and
intersecting thereal line. This a geometrically constructed
bijection between (0,1)and R.2- Use the Schroder-Bernstein
theorem: a) (0,1) in [0,1] in R b) R and (0,1) are equivalent
(by 1-)therefore all three sets are equivalent ([0,1], (0,1)
and R).
===
Subject: Re: 1 to 1 correspondence>How would I show
a 1 to 1 correspondence between (0,1) and [0,1] ?????Choose
any sequence of unique reals a_n from (0, 1). Letf: [0,1] ->
(0,1)be dened as0 |--> a_11 |--> a_2a_n |--> a_(n+2)a |--> 
a,
when a not in a_nThen f is a bijection.-- I'm not interested 
in
mathematics that might have anythingto do with reality. --
Russell Easterly, in sci.math
===
Subject: Re: 1 to 1
correspondence> How would I show a 1 to 1 correspondence
between (0,1) and [0,1] ????? One way is to dene f:[0,1] ->
(0,1) such that f(0) = 1/2,f(1) = 1/4,f(1/2^n) = 1/2^(n+2),
for all positive integers n,otherwise f(x) = x.The above is
bijective, but not order preserving. Order preserving and
bijective are mutually exclusive properties for mappings
between any closed interval of reals and any open interval of
reals.
===
Subject: Re: 1 to 1 correspondenceVirgil
to 1 correspondence between (0,1) and [0,1] ?????>> One
way is to dene f:[0,1] -> (0,1) such that> f(0) = 1/2,> f(1)
= 1/4,> f(1/2^n) = 1/2^(n+2), for all positive integers n,>
otherwise f(x) = x.>> The above is bijective, but not order
preserving.>> Order preserving and bijective are mutually
exclusive properties for> mappings between any closed interval
of reals and any open interval of> reals.Ok I see this.
===
Subject: Re: 1 to 1 correspondence> How would I show
a 1 to 1 correspondence between (0,1) and [0,1] ?????Hint: How
would you show a 1 to 1 correspondence
between{1,1/2,1/3,1/4,...} and {1/2,1/3,1/4,...}?-- Daniel W.
Johnsonpanoptes@iquest.nethttp://members.iquest.net/~panoptes/
039 53 36 N / 086 11 55 W
===
Subject: The infamous K-man
V|9[5IY]/OB15+<DlkA6*rsaeJb?9HnEE@[CAy1koT:N4Ew'*#35g{6z1>}/e
:
m!6.qw'Kzn@$#CJQl/UvR,&G?sNO{#LEEnS3lQRnT#Ql#'
3ySQ80}u7Q<`+B@;
Y;81AnK'_PD^fm.3T/<-i~YdZ%Q}b?:-]{Lr#s6.N,r(}Hi Peter ,
nnrp1.ozemail.com.au!53ab2750 .You wouldn't be the infamous
K-man by any chance, would you ?Sloppy, sloppy.
===
Subject: Re:
The infamous K-man ?> Hi Peter ,> 
>
nnrp1.ozemail.com.au!53ab2750 . > You wouldn't be the 
infamous
K-man by any chance,> would you ? Sloppy, sloppy.Jeff,Be
from ozemail.au (Australia).Double-A
===
Subject: Crikey mate
V|9[5IY]/OB15+<DlkA6*rsaeJb?9HnEE@[CAy1koT:N4Ew'*#35g{6z1>}/e
:
m!6.qw'Kzn@$#CJQl/UvR,&G?sNO{#LEEnS3lQRnT#Ql#'
3ySQ80}u7Q<`+B@;
Y;81AnK'_PD^fm.3T/<-i~YdZ%Q}b?:-]{Lr#s6.N,r(}Hi Double-A, 
You
from ozemail.au ( Australia ) . Crikey mate ! Maybe that is
him then.I think that's k-man's profession.
===
Subject:
V|9[5IY]/OB15+<DlkA6*rsaeJb?9HnEE@[CAy1koT:N4Ew'*#35g{6z1>}/e
:
m!6.qw'Kzn@$#CJQl/UvR,&G?sNO{#LEEnS3lQRnT#Ql#'
3ySQ80}u7Q<`+B@;
Y;81AnK'_PD^fm.3T/<-i~YdZ%Q}b?:-]{Lr#s6.N,r(}Oops, I called
ozemail.com.au!53ab2750 , an anonymous server ... It's 
not.So
what does the 53ab2750 mean ?It seems to be some kind of a
control code.
===
Subject: . 53ab2750 is an anonymous user
V|9[5IY]/OB15+<DlkA6*rsaeJb?9HnEE@[CAy1koT:N4Ew'*#35g{6z1>}/e
:
m!6.qw'Kzn@$#CJQl/UvR,&G?sNO{#LEEnS3lQRnT#Ql#'
3ySQ80}u7Q<`+B@;
Y;81AnK'_PD^fm.3T/<-i~YdZ%Q}b?:-]{Lr#s6.N,r(}I asked,
I
think it means that a proxy client is being used. ( i.e. The
poster is being kept anonymous )Normally that number would
identify the user.
===
Subject: Re: . 53ab2750 is an anonymous
<slrnc341gb.33p.dubious@radioactivex.lebesque-al.net> Jeff
Relf: >I asked,
 >I think it means that a proxy client is
being used. > ( i.e. The poster is being kept anonymous ) >
>Normally that number would identify the user. No, it
wouldn't. Go look up the rfc for the nntp protocol. The
pathprior to arriving on the news server on which you are
reading the
===
Subject: Re: . 53ab2750 is an anonymous user
V|9[5IY]/OB15+<DlkA6*rsaeJb?9HnEE@[CAy1koT:N4Ew'*#35g{6z1>}/e
:
m!6.qw'Kzn@$#CJQl/UvR,&G?sNO{#LEEnS3lQRnT#Ql#'
3ySQ80}u7Q<`+B@;
Y;81AnK'_PD^fm.3T/<-i~YdZ%Q}b?:-]{Lr#s6.N,r(}Hi Bilge,
You
mistakenly thought, No, it wouldn't. Go look up the rfc for
the nntp protocol. The path line identies the news servers
through which I know for a fact that a number like that 
 I
sometimes score on it, it works perfectly.For example it
identies one exact Earthlink user. take a look at Ha Ha
Hanson's headers,
 you would always know that it was him,
and only him.53ab2750 is an anonymous or unauthenticated
and anonymous posts ) .
===
Subject: Re: . 53ab2750 is an
<slrnc356kv.7o4.dubious@radioactivex.lebesque-al.net> Jeff
Relf: >Hi Bilge,
 >You mistakenly thought, > No, it
wouldn't. > Go look up the rfc for the nntp protocol. > The
path line identies the news servers through which >I know
for a fact that a number like that 
 > I sometimes score on
it, it works perfectly. No, you don'y know that for a fact,
for the same reason you don't knowanything else - you never
look up the references. To wit, RFC 1036 says:2.1.6. Path This
line shows the path the message took to reach the current
system. When a system forwards the message, it should add its
own name to the list of systems in the Path line. The names
may be separated by any punctuation character or characters
(except . which is considered part of the hostname). and
Normally, the rightmost name will be the name of the
originating system. However, it is also permissible to include
an extra entry on the right, which is the name of the sender.
This is for upward compatibility with older systems. The Path
line is not used for replies, and should not be taken as a
mailing address. It is intended to show the route the message
traveled to reach the local host. There are several uses for
this information. One is to monitor USENET routing for
performance... >For example it identies one exact Earthlink
user. > take a look at Ha Ha Hanson's headers,
 > you would
always know that it was him, and only him. Only because
earthlink _chooses_ to add that as an optional eldfor 
it's
own reasons. It has nothing to do with any requirementto do so
as rfc 1036 makes clear. As usual, you make invalid
assumptionsand draw unwarranted conclusions. Proof by example
dies not constitutea proof. >53ab2750 is an anonymous or
unauthenticated user. jeff relf is an anencephalic or
brainless poster.
===
Subject: Re: . 53ab2750 is an anonymous
Relf:>Hi Bilge,
> You mistakenly thought,> I know for
a fact that a number like that
> I sometimes score on it,
it works perfectly.> For example it identies one exact
Earthlink user.> take a look at Ha Ha Hanson's headers,
>
> you would always know that it was him, and only
him.>[Bilge]> jeff relf is an anencephalic or brainless
poster.>AHhahahahaha.......ahahahaha...AHAHAHAHHAA......Well,
Bilge, I don't know about your evaluation of Jeff,but I must
thank Jeff dearly that he takes such a deep andabiding
interest in my posted grand wisdoms and goes tothe extreme
length to verify and make sure that it is indeedfrom me,
hanson@quick.net = !ed99b2be
.........AHAHA....HAHAHA......ahahahahaha......AHAHAHHAHAHA...
...That's what I call readership loyalty, Bilge!Jeff, you 
get
three attaboys for this!AHahhahahaha....ahahahanson
===
Subject:
Re: . 53ab2750 is an anonymous user .hanson <hanson@quick.net>
> You
mistakenly thought,> I know for a fact that a number like
that
> I sometimes score on it, it works perfectly.>
For example it identies one exact Earthlink user.> take a
look at Ha Ha Hanson's headers,
> you would always know
that it was him, and only him.>> [Bilge]> jeff relf is an
anencephalic or brainless poster.>>
AHhahahahaha.......ahahahaha...AHAHAHAHHAA......> Well, Bilge,
I don't know about your evaluation of Jeff,> but I must 
thank
Jeff dearly that he takes such a deep and> abiding interest in
my posted grand wisdoms and goes to> the extreme length to
verify and make sure that it is indeed> from me,
hanson@quick.net = !ed99b2be .........AHAHA....>
HAHAHA......ahahahahaha......AHAHAHHAHAHA......> That's what 
I
call readership loyalty, Bilge!> Jeff, you get three attaboys
for this!> AHahhahahaha....ahahahansonIt sure doesn't take
much to make you laugh>
===
Subject: Re: . 53ab2750 is an
Jeff Relf:>Hi Bilge,
> You mistakenly thought,>
> I know for a fact that a number like that
> I
sometimes score on it, it works perfectly.> For example
it identies one exact Earthlink user.> take a look at
Ha Ha Hanson's headers,
> you would always know that
it was him, and only him.>> [Bilge]> jeff relf is an
anencephalic or brainless poster.>>
AHhahahahaha.......ahahahaha...AHAHAHAHHAA......> Well,
Bilge, I don't know about your evaluation of Jeff,> but I
must thank Jeff dearly that he takes such a deep and>
abiding interest in my posted grand wisdoms and goes to> the
extreme length to verify and make sure that it is indeed>
from me, hanson@quick.net = !ed99b2be .........AHAHA....>
HAHAHA......ahahahahaha......AHAHAHHAHAHA......> That's what
I call readership loyalty, Bilge!> Jeff, you get three
attaboys for this!> AHahhahahaha....ahahahanson It sure
doesn't take much to make you laugh>
>AHahahahaha....AHAHAhahahaha...ahahaha....Your are so right,
Peter. It does not take much to makeme smile, laugh, roar,
ROTL or ROTFLMAO...ahahahaha..That's why you too get three
attaboys for your reply !Life's a bowl of cherries, pits and
nuts and all....ahahahahaha.......ahahahahanson
===
Subject: Re:
53ab2750 ?> Oops, I called ozemail.com.au!53ab2750 ,> an
anonymous server ... It's not. So what does the 53ab2750 
mean
? It seems to be some kind of a control code.I just guessing
now...Could it be the IP address written in hexadecimal, i.e.
83.171.39.80?-Michael.
V|9[5IY]/OB15+<DlkA6*rsaeJb?9HnEE@[CAy1koT:N4Ew'*#35g{6z1>}/e
:
m!6.qw'Kzn@$#CJQl/UvR,&G?sNO{#LEEnS3lQRnT#Ql#'
3ySQ80}u7Q<`+B@;
Y;81AnK'_PD^fm.3T/<-i~YdZ%Q}b?:-]{Lr#s6.N,r(}Hi Michael
Jrgensen,
You ventured, Could it be the IP address ,A number
like 53ab2750 would normally identify the poster, but 53ab2750
seems to mean: An anonymous poster .
===
Subject: Re: wanted:
sha1:mfuAXgK49IwzXanM+hg4/lqFWMo=see.address@post.sig
these false proofs.> I'm surprised everybody here missed
this unique opportunity of killing two> birds with one
stone.>> Starting with a rock solid result from harrissian
fundamental tautology> space theory and developping at an
elementary level, here is a crystal clear> proof of a core
error in mathematics.> (0) 1 = 1>> (1) 1 = sqrt( 1 )>> (2) 1
= sqrt[ (-1)^2 ]>> (3) 1 = sqrt[ (-1) * (-1) ]>> (4) 1 = sqrt(
-1 ) * sqrt( -1 )>> (5) 1 = i * i>> (6) 1 = -1> Please note
that some nitpickers might argue that sqrt being such an>
ambiguous and lunatic operator, the odds of sqrt(-1)
evaluating to (i) the> rst time and (-i) the second time (or
vice-versa) are not completely> negligeable.You've missed 
the
linguistic turn this morning that explains thingsbetter now.
There are two numbers that, when squared, yield 1. Forthe
square root operator to *always* return the positive of these
twowould require human *intent*. Mathematics always works for
a reason,but humans choose arbitrarily and so while they can
form intent, QueenMathematics cannot.Therefore, sometimes the
square root operator must sometimes choosethe negative root.--
Jesse F. Hughes[Mathematical] society has evolved far enough
away from mainstreamsociety that it has become rogue, and now
is willing to push its needsagainst that of the majority. --
James S. Harris 
===
Subject: New crypto algorithmDedicated to
the Faculty of Mechanics and Mathematics of Belarusian
StateUniversityThis message denes new symmetric
cryptoalgorithm and contains animplementation that can be used
for practical testing.The cryptoalgorithm has the following
advantages: 1. Keys of arbitrary length are allowed. 2. The
cryptoalgorithm is extremely easy to understand and implement.
3. The cryptoalgorithm is strong (not proved), open and free. I
am theoriginator of this cryptoalgorithm and I tell you: use it
for free.The cryptoalgorithm is based on the process of
conversion from one numericsystem to another: a number is
converted to another numeric system, thedigits are permuted
and the result is converted back to the original
numericsystem.Description of the cryptoalgorithm:- Plain text
to encode is (long) unsigned integer number. Let's denote it
byA.- Similarly, key is unsigned integer number B. 0. Plain
text is compressed in order to transform it into a relatively
random(i.e. uncompressible) sequence of bytes using some
compression algorithm. 1. A is split into a sequence of
remainders of division by B. 2. The sequence of remainders is
permuted. B's value denes the permutation. 3. 
The permuted
sequence is converted to number A2.- A2 is the encoded text.
The following Java program is created for practical testing.
It is a completeimplementation of the cryptoalgorithm (without
compression phase).-------------------------------- A.java
--------------------------------import java.io.File;import
java.io.FileInputStream;import java.io.FileOutputStream;import
java.math.BigInteger;import java.util.LinkedList;import
java.util.ListIterator;public class A{ public static void
main(String[] args) throws Exception { if (args.length!=4)
badArgs(); String scmd=args[0]; String skey=args[1]; String
sin=args[2]; String sout=args[3]; if ( !(e.equals(scmd) ||
d.equals(scmd)) ) badArgs(); boolean encrypt=e.equals(scmd);
BigInteger b=new BigInteger(1, readFile(skey, 0)); if
(encrypt) { byte[] adata=readFile(sin, 1); // append positive
most signicant (random) byte for (int i=1; i<adata.length;
i++) adata[0]^=adata[i]; adata[0]&=(byte)0x7f; // clear
10000000 bit adata[0]|=(byte)0x40; // set 01000000 bit
BigInteger a=new BigInteger(adata); BigInteger[]
rems=computeRemainders(a, b); BigInteger[] rems2=permute(rems,
b, false); BigInteger a2=computeNumber(rems2, b);
writeFile(sout, a2.toByteArray(), 0); } else { BigInteger
a=new BigInteger(readFile(sin, 0)); BigInteger[]
rems=computeRemainders(a, b); BigInteger[] rems2=permute(rems,
b, true); BigInteger a2=computeNumber(rems2, b);
writeFile(sout, a2.toByteArray(), 1); } } private static void
badArgs() { System.err.println(Invalid args. Must be:);
System.err.println(e|d key_le input_le 
output_le);
System.exit(1); } private static byte[] readFile(String fname,
int off) throws Exception { File f=new File(fname);
FileInputStream n=new FileInputStream(f); try { int
len=(int)f.length(); byte[] ret=new byte[len+off];
n.read(ret, off, len); return ret; } nally { 
n.close(); } }
private static void writeFile(String fname, byte[] data, int
off) throws Exception { FileOutputStream fout=new
FileOutputStream(fname); try { fout.write(data, off,
data.length-off); } nally { fout.close(); } } public static
BigInteger[] computeRemainders(BigInteger a, BigInteger b) {
LinkedList lst=new LinkedList(); while
(!a.equals(BigInteger.ZERO)) { BigInteger[]
qr=a.divideAndRemainder(b); lst.add(qr[1]); a=qr[0]; } return
(BigInteger[])lst.toArray(new BigInteger[lst.size()]); }
public static BigInteger computeNumber(BigInteger[] rems,
BigInteger b) { BigInteger a=BigInteger.ZERO; for (int
i=rems.length-1; i>=0; i--) { a=a.multiply(b);
a=a.add(rems[i]); } return a; } public static BigInteger[]
permute(BigInteger[] rems, BigInteger b, boolean inverse) {
int len=rems.length; BigInteger[] ret=new BigInteger[len]; int
pos[]=createPermutation(len, b, inverse); for (int i=0; i<len;
i++) ret[pos[i]]=rems[i]; return ret; } public static int[]
createPermutation(int len, BigInteger b, boolean inverse) {
int[] ret=new int[len]; LinkedList lst=new LinkedList(); for
(int i=0; i<len; i++) lst.add(new Integer(i)); BigInteger
b2=b; for (int div=len; div>=1; div--) { BigInteger bdiv=new
BigInteger(+div); if (div==len) // always move
bdiv=bdiv.subtract(BigInteger.ONE); // most signicant
remainder if (b2.compareTo(bdiv)<0) b2=b; // restart from b
BigInteger[] qr=b2.divideAndRemainder(bdiv); b2=qr[0]; int
rem=qr[1].intValue(); ListIterator lit=lst.listIterator(rem);
ret[div-1]=((Integer)lit.next()).intValue(); lit.remove(); }
if (inverse) { int[] ret2=new int[len]; for (int i=0; i<len;
i++) ret2[ret[i]]=i; return ret2; } return ret;
}}-----------------------------------8<-----------------------
------------Type javac A.javato compile the source. A.class
le will appear.To encode input_le using 
key_le, type java A
e key_le input_le output_leThe 
encoded text will be saved in
output_le.To decode output_le using 
key_le, type java A d
key_le output_le input_le2The 
decoded text will be saved in
input_le2 that must be equal to theoriginal
input_le.Implementation notes: 0. The compression phase
increases the strength of the cryptoalgorithm. Itdecreases
probability of getting duplicate remainders, that is it
increasesthe permutation quality. In practice, duplicate
remainders are very rarebecause their number substantively
less than B's value. The compressionalgorithm does not have 
to
obey any special requirements, ordinary Huffmancoding ts 
well.
1. The permutation algorithm must be based on key and choose a
permutation inequiprobable way. And it must always move the
most signicant remainderbecause leading bytes of the encoded
text must differ from the original plaintext. The used
algorithm is rather straightforward and can probably
beimproved. 2. Obviously, value of a number does not depend on
leading zeros but leadingzeros must not be lost during
encoding/decoding of A. That's why before theencoding, some
non-zero byte is appended to A. This byte will be
discardedbefore the writing of decoded text. 3. BigInteger
datatype is signed and the most signicant bit 
denes thesign.
As long as the cryptoalgorithm operates with unsigned integers
thespecial BigInteger constructor is used to create B's 
value.
The constructorallows to explicitly set the sign.
Unfortunately, this constructor can appendleading zero byte so
it is not used for A. Instead, the most signicant bit(of the
byte that is appended to A during construction) is always
cleared. 4. The values of the minor bits of the byte that is
appended to A arecomputed via hashing bytes of the plain text.
Actually, any number of byteswith any values can be appended to
A. For example, their values can depend onthe current system
time. In this case, the same key and plain text will leadto
different encoded texts. This modication of the
cryptoalgorithmsubstantively increases its
strength.Recommendations: 1. The key must consist of a random
sequence of bits and be not less than 32bytes (i.e. 256 bits).
2. Uneven (i.e. the least signicant bit of last byte of the
le is 1) keysshould be used. It ensures that 2 and B are
relatively prime numbers (have nocommon divisors). 3. To
increase the strength of the cryptoalgorithm, plain text
le(input_le) must be at least 10 times longer 
than key_le.
Obviously, input_le is shorter than 
key_le no encryption is
possible. 4. As long as encryption/decryption time
substantively depends on lengths ofles, it makes sense to
split plain text le into parts of aboutkey_length*1000 size
(i.e. 32000 bytes for 32 bytes key). Also it is not sowise to
use keys that longer than 64 bytes (i.e. 512 bits) in
practice.Notes: 1. All I know about this cryptoalgorithm is
described above. I have _nothing_to add. 2. If you have some
question but thoughtful analysis of the description andthe
implementation does not lead to the answer then I do not know
the answertoo. And I will not be able to help, sorry. 3. If
you have discovered a mathematical ¤aw or found a bug in
theimplementation, please mail me to
new_crypto_algorithm@yahoo.com. The messagemust be concise and
have a descriptive subject. All other mails will beignored (no
===
Subject:
Re: Metamath Axiom of Choice> D.McAnally@i'm_a_gnu.uq.net.au
specicity, it is a>>wff of the FOL of set theory which is
(roughly speaking) stipulated>>true, and can thus be used in
deductions. If you decide that you>>don't like the axiom of
choice, you can choose to not work with FO>>languages which
include it. But even then, it's hard to deny 
it's use>>in
axiom systems which _do_ use it. (Since, by construction,
you're>>setting up a conditional... If the axiom of choice
is true, and>>...blah..., then ...blah'...)>>In any
case, there is nothing to prove.>As I stated above, the
Metamath formulation of the Axiom of Choice is >effectively
the statement that given any set x, there exists a set y >
>such that for all nonempty elements w of x, w n (Us) has
exactly one >element, where s = {t in y : w in t}. This is
not amongst the >formulations of the Axiom of Choice of
which I was previously aware.>The question of the
equivalence of the Metamath formulation with the>more
standard versions under ZF naturally arises, as such an
equivalence >is necessary if the Metamath set theoretic
axioms are to be taken as >axioms of ZFC. Also, the
motivation behind proving the equivalence of the Metamath >
formulation of the Axiom of Choice and more standard
formulations> is the same as the motivation behind proving the
equivalence of the > Axiom of Choice and Zorn's Lemma, and 
the
equivalence of the Axiom > of Choice and the Well-Ordering
Theorem. I presume that you have > seen the proofs of these
last two mentioned equivalences, Acid Pooh.> Heh. Since you
appear to be goading me into replying, and I'm asucker for
this sort of thing, here goes: I was not aware thatMetamath
was a website. Nor was I aware that they had a
differentversion of the axiom of choice. I assumed (obviously
incorrectly)that you meant metamath to refer to Set Theory, or
some suchmetamathematical study. Now, in this context, my
response doesn'tseem so inappropriate, does it? In 
particular,
it shouldn't seem sopatronizing, since, as I parsed your
question, it is a question abeginner should ask. So, why
decide to patronize me?'cid .95oohPS. Chill out.
===
Subject: Re:
Metamath Axiom of Choice>> D.McAnally@i'm_a_gnu.uq.net.au
formulation of the Axiom of Choice is >>effectively the
statement that given any set x, there exists a set y >>such
that for all nonempty elements w of x, w n (Us) has exactly
one >>element, where s = {t in y : w in t}. As stated, it is
false. Add the elements of x are disjointand it is one of the
standard forms.To see that it is false as stated, let x have
three elements, {1, 2}, {1, 3}, {2, 3}. Any set intersecting
all of these inat least one element has to contain both
elements of one.-- This address is for information only. I do
not claim that these viewsare those of the Statistics
Department or of Purdue University.Herman Rubin, Department of
Statistics, Purdue Universityhrubin@stat.purdue.edu Phone:
(765)494-6054 FAX: (765)494-0558
===
Subject: Re: Metamath Axiom
...................>As I stated above, the Metamath
formulation of the Axiom of Choice is >effectively the
statement that given any set x, there exists a set y >such
that for all nonempty elements w of x, w n (Us) has exactly one
>element, where s = {t in y : w in t}. >As stated, it is
false. Add the elements of x are disjoint>and it is one of the
standard forms.>To see that it is false as stated, let x have
three elements, >{1, 2}, {1, 3}, {2, 3}. Any set intersecting
all of these in>at least one element has to contain both
elements of one.But in the statement above, s is explicitly
t}), so the claim is not the existence of a single set with one
element in commonwith every nonempty element.If we take x =
{{1,2},{1,3},{2,3}}, then take y =
{{{1,2},1},{{1,3},1},{{2,3},2}}.For the element w = {1,2}, s =
{t in y : w in t} = {{{1,2},1}}, so that Us = {{1,2},1}, and so
w n (Us) = {1} has exactly one element.For the element w =
{1,3}, s = {t in y : w in t} = {{{1,3},1}},so that Us =
{{1,3},1}, and so w n (Us) = {1} has exactly oneelement.For
the element w = {2,3}, s = {t in y : w in t} = {{{2,3},2}},so
that Us = {{2,3},2}, and so w n (Us) = {2} has exactly
oneelement.David McAnally At the moment, they (the Time Lords)
are far from being all-powerful. That's why 
it's been left up
to me and me and me. quote by: Patrick Troughton in The Three
Doctors-------
===
Subject: Re: Metamath Axiom of Choice
<c0me61$d3t$1@bunyip.cc.uq.edu.au>
<c0mke8$j39$1@bunyip.cc.uq.edu.au>
sha1:7KR/2dC3CE+vTR0AUl1MPgWQp3U=poohonlsd@yahoo.com (Acid
replying, and I'm a> sucker for this sort of thing, here 
goes:
I was not aware that> Metamath was a website. Nor was I aware
that they had a different> version of the axiom of choice. I
assumed (obviously incorrectly)> that you meant metamath to
refer to Set Theory, or some such> metamathematical study.
Now, in this context, my response doesn't> seem so
inappropriate, does it? In particular, it shouldn't seem so>
patronizing, since, as I parsed your question, it is a
question a> beginner should ask. So, why decide to patronize
me?Perhaps he decided to patronize you because, as *everyone*
parsed yourpost, you are incapable of reading plain English
but you pretend tohave the expertise to respond to questions
anyway?It shouldn't seem patronizing after all, since as he
(and otherrespondents) read your post, it was an answer an
ignoramus mightwrite.-- You lack the ability to reason, but
instead get an idea in your headand hold on to it against all
evidence. I don'tfind you credible,and reject 
your claims, as
coming from a ¤awed source. -- James S Harris shoots for
===
Subject: Re: Metamath
Perhaps he decided to patronize you because, as *everyone*
parsed your> post, you are incapable of reading plain English
but you pretend to> have the expertise to respond to questions
anyway?Well, I guess you must be right, then. Since *everyone*
says so. Bythe way, you should note that what I said was
_accurate_. I onlyresponded to the wrong question. I really
don't see why my gaffe hasinspired you to post two insulting
messages about me. It shouldn't seem patronizing after all,
since as he (and other> respondents) read your post, it was an
answer an ignoramus might> write.I'm not exactly sure why 
you'd
want to get into a pissing contest withme. But your reply is
completely pointless, since it is in responseto a post that
wasn't addressed to you.'cid 
.95oohPS. Chill out. This has
nothing to do with you.
===
Subject: Re: Metamath Axiom of
Choice>> Also, the motivation behind proving the equivalence
of the Metamath >> formulation of the Axiom of Choice and more
standard formulations>> is the same as the motivation behind
proving the equivalence of the >> Axiom of Choice and Zorn's
Lemma, and the equivalence of the Axiom >> of Choice and the
Well-Ordering Theorem. I presume that you have >> seen the
proofs of these last two mentioned equivalences, Acid Pooh.>
Heh. Since you appear to be goading me into replying, and 
I'm
a> sucker for this sort of thing, here goes: I was not aware
that> Metamath was a website. Nor was I aware that they had a
different> version of the axiom of choice. I assumed
(obviously incorrectly)> that you meant metamath to refer to
Set Theory, or some such> metamathematical study. Now, in this
context, my response doesn't> seem so inappropriate, does 
it?
In particular, it shouldn't seem so> patronizing, since, as 
I
parsed your question, it is a question a> beginner should ask.
So, why decide to patronize me?I was likewise unaware that
there was a Metamath site, but that factbecame abundantly
clear as soon as I read D. McAnally's very 
rstsentence of his
rst post on the subject. I suggest you go back andread it.--
Dave SeamanJudge Yohn's mistakes revealed in Mumia Abu-Jamal
ruling.<http://www.commoncouragepress.com/index.cfm?action=
book&bookid=228>
===
Subject: Re: Metamath Axiom of Choice>>
Also, the motivation behind proving the equivalence of the
Metamath >> formulation of the Axiom of Choice and more
standard formulations>> is the same as the motivation behind
proving the equivalence of the >> Axiom of Choice and Zorn's
Lemma, and the equivalence of the Axiom >> of Choice and the
Well-Ordering Theorem. I presume that you have >> seen the
proofs of these last two mentioned equivalences, Acid Pooh.>
> Heh. Since you appear to be goading me into replying, and
I'm a> sucker for this sort of thing, here goes: I was not
aware that> Metamath was a website. Nor was I aware that
they had a different> version of the axiom of choice. I
assumed (obviously incorrectly)> that you meant metamath to
refer to Set Theory, or some such> metamathematical study.
Now, in this context, my response doesn't> seem so
inappropriate, does it? In particular, it shouldn't seem so>
patronizing, since, as I parsed your question, it is a question
a> beginner should ask. So, why decide to patronize me? I was
likewise unaware that there was a Metamath site, but that fact>
became abundantly clear as soon as I read D. McAnally's very
rst> sentence of his rst post on the subject. 
I suggest you
go back and> read it.It's quite obvious that 
I'm aware of it
now. Your suggestion isappreciated, but you seem to have
suffered the same mistake as me. :)'cid .95ooh
===
Subject: Re:
Metamath Axiom of Choice> poohonlsd@yahoo.com (Acid Pooh)
specically at their statement of>>the Axiom of Choice, I
note that their statement of the Axiom of Choice is>>
>effectively the statement that for any set x, there exists a
set y such>>that for any nonempty element w of x, there is
at least one element t of y>>such that w is an element of
t, and that w has exactly one element in>>common with the
union of all elements of y which have w as an element,>>
>i.e. there is a set y such that w is an element of some
element t of y,>>and if s = {t in y : w in t}, then w n
(Us) has exactly one element, where>>A n B denotes the
intersection of A and B, and where Us is the union of>>the
elements of s.>>That this implies the Axiom of Choice is
obvious (if w is a nonempty >>element of x, let f(w) be
the unique element of w n (Us), where >>s = {t in y : w in
t}). The question is how can one prove the Metamath>>
>statement of the Axiom of Choice in ZFC?>> How does this
sound for a proof. Let f : x-{empty set} -> Ux be a choice >
>> function (so that f(v) is an element of v for all nonempty
elements v of x). >> Let y = {{v,f(v)} : v in x, v
nonempty}. The rst thing to note is that >> if w is a
nonempty element of x, then there is an element of y which has
w >> as an element, specically, {w,f(w)}. Let w be a
nonempty element of x, >> then s = {t in y : w in t} =
{{w,f(w)}} u {{v,w} : v in x, v nonempty, >> f(v) = w},
where A u B denotes the union of A and B. It follows that >>
Us = {w,f(w)} u {v in x : v nonempty, f(v) = w}, and so the
elements of >> Us are w, f(w), and nonempty elements v of x
such that f(v) = w. Since >> w is an element of v for all
nonempty elements v of x such that f(v) = w, >> then the
elements of Us are w, f(w) and some sets which have w as an >
>> element. By the axiom of regularity, the only common
element of w and Us >> is f(w), completing the proof of the
Metamath statement of the Axiom of >> Choice.>> David
McAnally>> At the moment, they (the Time Lords) are far
from being all-powerful.>> That's why it's 
been left up to
me and me and me.>> quote by: Patrick Troughton in The Three
Doctors>> ------->Uhm... the Axiom of Choice is an
_axiom_. For specicity, it is a>wff of the FOL of set
theory which is (roughly speaking) stipulated>true, and can
thus be used in deductions. If you decide that you>don't
like the axiom of choice, you can choose to not work with FO>
>languages which include it. But even then, it's hard to 
deny
it's use>in axiom systems which _do_ use it. (Since, by
construction, you're>setting up a conditional... If the
axiom of choice is true, and>...blah..., then ...blah'...)>
>In any case, there is nothing to prove. As I stated above,
the Metamath formulation of the Axiom of Choice is >
effectively the statement that given any set x, there exists a
set y > such that for all nonempty elements w of x, w n (Us)
has exactly one > element, where s = {t in y : w in t}. This
is not amongst the > formulations of the Axiom of Choice of
which I was previously aware.> The question of the equivalence
of the Metamath formulation with the> more standard versions
under ZF naturally arises, as such an equivalence > is
necessary if the Metamath set theoretic axioms are to be taken
as > axioms of ZFC. In spite of what you might think, I was NOT
trying to prove the Axiom of> Choice from ZF. If you go back
initially gave a proof of a more standard version of the
Axiom> of Choice from ZF *and* the Metamath version of the
Axiom of Choice, and> that I was wondering about the
possibility of a proof of the Metamath> version of the Axiom
of Choice from ZFC (which I believe that I supplied> in the
subsequent posting). In fact, I can't tell where you could
have> got any idea that I was trying to prove the Axiom of
Choice from ZF. Your condescendingly pointing out that the
Axiom of Choice is an axiom,> and your nal statement that
there is nothing to prove demonstrate that> you have
completely missed the point of what I was trying to do, i.e.
to> demonstrate that the axioms adopted by Metamath do in fact
form a set of> axioms for ZFC, a fact which was not immediately
obvious to me. Both Jesse Hughes and David Ullrich picked up
what I was trying to do. Why> couldn't you?> My apologies 
for
not reading your post more carefully.I'll have to be more
careful in the future, so as not to offend thoseasking for
help with my condescending remarks.'cid .95oohPS. 
Chill
out.
===
Subject: Re: branch of log z> A veil has been woven out
of words like imaginary and spread over> the real plane, giving
the impression of an domain, accessible only> for people with
special brain-power, sometimes letting the real plane>
shimmering through.> introduced to me (at the college level,
anyways). > The rst time i hear this, except from Caspar
for me, and i'm> sincerely Yours for these words.> No. 
It's
known to insiders, working under the veil - but it's not 
made>
well known.That was a really long post, and you snipped the
part where Imentioned that both approaches are useful. Now,
quite obviously, whatI meant is that this is well known to
people who are interested inmathematics. Nobody else really
cares that the complex plane is justR^2. Nobody else cares
that the complex eld is R^2 with some neatmultiplication
thrown in. Call us insiders if you'd like... butapparently
you're interested, too. So stop throwing stones, 
we'reboth in
the glass house. Can You give me a reference, where it's 
said,
that the i-axis and the> y-axis are just different names, but
with no mathematical difference?> The same for R2, the
2D-plane, no difference to the complex plane,> argand
diagramm, (gauss plane)?R^2 and the Complex plane are
isomorphic if you *ignore* complexmultiplication. That is
about as deep as the relationship goes. (It's quite a deep
relationship, actually. But less deep than youapparently
think)> Can You give me a reference, where
mixed-mode-calculating is used,> like :> i*(3,4)=(-4,3) (see
my calculator)?> Can You name someone, who uses the notation
3+i*4 in any> 2D-vectorspace in place of (3,4) ? The second is
of course an ordered> pair of reals and the rst is using the
linear-combination of two> basis-vectors (1,0) and i=(0,1)
and, as the real numbers are embedded,> you ommit (1,0).>
That's terrible notation. The i notation on elements from R 
is
muchnicer. And there is no reason to use the notation a + bi
when workingwith R^2, because, in particular, multiplication
is not dened onR^2. So, if we want to generalize to R^n, do
we need n-1 letters torepresent the n directions?> And
sometimes insiders entangle in their own veil and can't see
clear,> look at Algebra: What is the relation between
(R2,+,r.s.m.) and> (R2,+,*)-this is called the vectorspace of
complex numbers(and an> commutative eld)- and
(R2,+,r.s.m.,dot)-this is the euclidian> vectorspace? branches
of log ...This stuff can get messy later. -> but may be without
imaginary veil a little bit less.Read Rudin's principles of
mathematical analysis. Or J.E. Marsden'sComplex Analysis. Or
Royden's analysis book. Or...'cid 
.95ooh
===
Subject: Re: We come
from your futureTime to commit suicide...
===
Subject: Re: A
question about Kripke semantics and physics (Was: Re: Study
groups in science): So, once again, let me start with a quick
Google search to: establish context.That is always a fun way
to learn, and getting better every year!: If I use the search
string .9526 dimensions string physics' I get 
about959: hits.::
For my part, I could care less about the details. It is the
mathematics: through which your colleagues express their
explanations that is importantto: me. The 26 dimensions are
particularly interesting here because their: symmetries and
invariants involved offer me an opportunity to ask you howto:
think about truth and falsity in physical theory.:: Now,
Galathaea has been wanting to talk about Heyting algebras and
quantum: logic. Section 6.4 of the paper::
http://www.illc.uva.nl/Publications/ResearchReports/MoL-2001-
09.text.pdf:: is entitled Finite projective formulas in two
variables Strangelyenough,: there are precisely 26 Heyting
algebras associated with 2-universal models: discussed here.
Moreover, there is not a single mention of quantumlogic.::
Now, isn't this just an amazing coincidence? Logicians
tellmathematicians: about .95T' and 
.95F,' the physicists are
talking about 26 dimensions, andmitch: knows just where to 
nd
a paper specifying the 26 2-generated Heyting: algebras.::
Actually, I do not think it is coincidental. Unfortunately,
whereas Iwould: love to offer an explanation, you would simply
engage in more vulgarity.I only today got through this
reference in my rst run through and stillhave much 
organising
to get a better understanding, but I was wondering ifyou were
thinking of a particular relationship here between the
constraintequation in string theory and the n=2 formula in the
paper. The constraintfor strings arises from a requirement for
Weyl invariance of the action,which has a very interesting
topological structure, so I can certainly see amotif. Have you
pursued this to the next exposition stage, or are you
stillhunting down the connections? This paper is very
interesting!=)--
-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-galathaea:
prankster, fablist, magician, liar
===
Subject: Understanding
understand
the taylor exapnsion for the sinefunction,sin(x) = x - x^3/3!
+ x^5/5! - x^7/7! + ...I've been told that it uses the fact
that e^(i*x) = cos(x) + i*sin(x),and I can easily work out the
taylor series for e^x, but I'm not surehow to use these two
pieces tofind the expansion for sin(x).Any help would be
===
Subject: Re:
trying to understand the taylor exapnsion for the sine>
function, sin(x) = x - x^3/3! + x^5/5! - x^7/7! + ... I've
been told that it uses the fact that e^(i*x) = cos(x) +
i*sin(x),> and I can easily work out the taylor series for
e^x, but I'm not sure> how to use these two pieces to 
nd the
Jonathan ChristensenDo you know, or know how tofind, the
expansion for e^x?If so, replace the x by ix, and see what
happens.
===
Subject: Re: Understanding taylor expansion for
sine> I've been trying to understand the taylor exapnsion 
for
the sine> function, sin(x) = x - x^3/3! + x^5/5! - x^7/7! +
... I've been told that it uses the fact that e^(i*x) = 
cos(x)
+ i*sin(x),> and I can easily work out the taylor series for
e^x, but I'm not sure> how to use these two pieces to 
nd the
expansion for sin(x).Hello.I usually work out the Taylor
series for sine around zero directly.Since you say you can
easily work out the Taylor series for e^(i*x)directly, I 
won't
go further on that point.If one plugs (i*x) into the Taylor
series for e^x around zero andtakes the imaginary terms alone,
doesn't one end up with the Taylorseries for sine around
zero?-- Try
http://csf.colorado.edu/pkt/pktauthors/Vienneau.Robert/
Bukharin.html To solve Linear Programs: .../LPSolver.htmlr c A
game: .../Keynes.html v s a Whether strength of body or of
mind, or wisdom, or i m p virtue, are found in proportion to
the power or wealth e a e of a man is a question t perhaps 
to
be discussed by n e . slaves in the hearing of their masters,
but highly @ r c m unbecoming to reasonable and free men in
search of d o the truth. -- Rousseau
===
Subject: Re:
-0800, dzhonatan@hotmail.com (Jonathan>I've been trying to
understand the taylor exapnsion for the sine>function,>>sin(x)
= x - x^3/3! + x^5/5! - x^7/7! + ...>>I've been told that it
uses the fact that e^(i*x) = cos(x) + i*sin(x),>and I can
easily work out the taylor series for e^x, but I'm not
sure>how to use these two pieces tofind the expansion for
sin(x). e^(ix) = cos(x) + i sin(x)<=> -i e^(ix) = -i cos(x) +
sin(x) => sin(x) = Re(-i e^(ix))Work from there.-- I'm not
interested in mathematics that might have anythingto do with
reality. -- Russell Easterly, in sci.math
===
Subject: Re:
Understanding taylor expansion for sinejust pick out the bits
that has an is in.Jonathan Christensen <dzhonatan@hotmail.com>
the
taylor exapnsion for the sine> function,>> sin(x) = x - x^3/3!
+ x^5/5! - x^7/7! + ...>> I've been told that it uses the 
fact
that e^(i*x) = cos(x) + i*sin(x),> and I can easily work out
the taylor series for e^x, but I'm not sure> how to use 
these
two pieces tofind the expansion for sin(x).>> Any help would
Understanding taylor expansion for sine
 > e^(i*x) = cos(x)
+ i*sin(x)thus sin(x) is the imaginary part of e^(ix), so you
have to extract the imaginary part of the series.
===
Subject:
UnionsLet A1 ,A2, ... be a sequence of subsets of a universal
set X. Dene Ej =( Aj - {A1 U A2 U...U A(j-1)} ) where j=
1,2,3,....I was able to prove that Ei intersect = Ej empty set
if i =/ j. I was alsoable to show that the union of Ek from k =
1 to n = union from k = 1 to n ofAk but I'm not sure how to
prove this last statement if k goes from 1 to oo.
===
Subject:
Re: Unions
===
Subject: Unions >Let A1 ,A2, ... be a sequence of
subsets of a universal set X. >Dene Ej = ( Aj - {A1 U A2 
U...U
A(j-1)} ) where j= 1,2,3,.... >I was able to prove that Ei
intersect Ej = empty set if i =/ j. >union of Ek from k = 1 to
n = union from k = 1 to n of Ak >how to prove this last
statement if k goes from 1 to oo. /{ Ej | j in N } = /{ Aj | j
in N }/{ Ej | j in N } subset /{ Aj | j in N } is easyConversly
assume x in /{ Aj | j in N } some k in N with x in Ak x in /{
Aj | j <= k } x in /{ Ej | j <= k } x in /{ Ej | j in N
}----
===
Subject: countable setsLet E1, E2, E3, ... be a
sequence of pairwise disjoint countable sets. Provethat union
of Ek as k = 1 to oo is countable. I do not need help
provingthis. I need someone to explain exactly what is being
asked. In Let E1, E2,E3,... be a sequence of pairwise disjoint
countable sets is that sayingthat the numbers of Ei's are
countable or is it saying that each Ei has acountable number
of elements??
===
Subject: Re: countable sets> Let E1, E2, E3,
... be a sequence of pairwise disjoint countable sets. Prove>
that union of Ek as k = 1 to oo is countable. I do not need
help proving> this. I need someone to explain exactly what is
being asked. In Let E1, E2,> E3,... be a sequence of pairwise
disjoint countable sets is that saying> that the numbers of
Ei's are countable or is it saying that each Ei has a>
countable number of elements?? Both.
===
Subject: Re: countable
sets> Let E1, E2, E3, ... be a sequence of pairwise disjoint
countable sets. Prove> that union of Ek as k = 1 to oo is
countable. I do not need help proving> this. I need someone to
explain exactly what is being asked. In Let E1, E2,> E3,... be
a sequence of pairwise disjoint countable sets is that saying>
that the numbers of Ei's are countable or is it saying that
each Ei has a> countable number of elements??Others have
explained what is being asked, but I was intrigued by
yourstatement that you don't need help proving it. Now that
the question hasbeen claried, you might look again at 
proving
it.Hint: If your proof does not invoke the axiom of choice,
then your proos wrong.-- Dave SeamanJudge 
Yohn's mistakes
revealed in Mumia Abu-Jamal
ruling.<http://www.commoncouragepress.com/index.cfm?action=
book&bookid=228>
===
Subject: Re: countable setsDave Seaman
be a sequence of pairwise disjoint countable sets.Prove> that
union of Ek as k = 1 to oo is countable. I do not need help
proving> this. I need someone to explain exactly what is
being asked. In Let E1,E2,> E3,... be a sequence of pairwise
disjoint countable sets is thatsaying> that the numbers of
Ei's are countable or is it saying that each Ei hasa>
countable number of elements??>> Others have explained what is
being asked, but I was intrigued by your> statement that you
don't need help proving it. Now that the question has> been
claried, you might look again at proving it.Dave,Now that I
understand the statement I will try to prove it using the
axiomof choice.Onto another topic. I hope that you don't 
mind.
I know that Mumia is notfacing a death sentence at this point
but is he still physically on deathrow locked up 23 hours per
day? FREE Mumia now!>> Hint: If your proof does not invoke the
axiom of choice, then your proof> is wrong.>> --> Dave
Seaman> Judge Yohn's mistakes revealed in Mumia Abu-Jamal
ruling.>
<http://www.commoncouragepress.com/index.cfm?action=book&
bookid=228>
===
Subject: OT (Re: countable sets)> Dave Seaman
hope that you don't mind. I know that Mumia is not> facing a
death sentence at this point but is he still physically on
death> row locked up 23 hours per day? FREE Mumia now!Neither
side was satised with Judge Yohn's decision 
and both sides
areappealing. The possibility still exists that the death
penalty could bereinstated without a new trial.-- Dave
SeamanJudge Yohn's mistakes revealed in Mumia Abu-Jamal
ruling.<http://www.commoncouragepress.com/index.cfm?action=
book&bookid=228>
===
Subject: Re: countable sets>> Let E1, E2,
E3, ... be a sequence of pairwise disjoint countable sets.
Prove>> that union of Ek as k = 1 to oo is countable. I do not
need help proving>> this. I need someone to explain exactly
what is being asked. In Let E1, E2,>> E3,... be a sequence of
pairwise disjoint countable sets is that saying>> that the
numbers of Ei's are countable or is it saying that each Ei 
has
a>> countable number of elements?? Others have explained what
is being asked, but I was intrigued by your> statement that
you don't need help proving it. Now that the question has>
been claried, you might look again at proving it. Hint: If
your proof does not invoke the axiom of choice, then your
proof> is wrong.Explicitely invoked? j gWhat about N ---> N x
N ---> union(E_k)where j is a suitable bijection and g(k,n) :=
f_k(n)is dened via given surjective maps f_k : N ---> Ek
?Marc
===
Subject: Re: countable sets> Let E1, E2, E3, ... be
a sequence of pairwise disjoint countable sets. Prove> that
union of Ek as k = 1 to oo is countable. I do not need help
proving> this. I need someone to explain exactly what is
being asked. In Let E1, E2,> E3,... be a sequence of
pairwise disjoint countable sets is that saying> that the
numbers of Ei's are countable or is it saying that each Ei 
has
a> countable number of elements??>> Others have explained
what is being asked, but I was intrigued by your>> statement
that you don't need help proving it. Now that the question
has>> been claried, you might look again at proving it.>>
Hint: If your proof does not invoke the axiom of choice, then
your proof>> is wrong.> Explicitely invoked? > j g> What about
N ---> N x N ---> union(E_k)> where j is a suitable bijection
and g(k,n) := f_k(n)> is dened via given surjective maps f_k
: N ---> Ek ?For any given k we are given that there is a
suitable surjection f_k : N-> E_k. That is, the set S_k of
surjections from N onto E_k is nonemptyfor each k.How do you
conclude (without AC) that there is a g: N x N ->
union(E_k)?That requires choosing all the f_k's at once, and
there are innitelymany of them. That is, you need to show
that there is a choice functionon the set { S_k : k in N }.--
Dave SeamanJudge Yohn's mistakes revealed in Mumia Abu-Jamal
ruling.<http://www.commoncouragepress.com/index.cfm?action=
book&bookid=228>
===
Subject: Re: countable sets* Math Tutor>
Let E1, E2, E3, ... be a sequence of pairwise disjoint
countable sets. Prove> that union of Ek as k = 1 to oo is
countable. I do not need help proving> this. I need someone to
explain exactly what is being asked. In Let E1, E2,> E3,... be
a sequence of pairwise disjoint countable sets is that saying>
that the numbers of Ei's are countable or is it saying that
each Ei has a> countable number of elements??... a sequence of
pairwise disjount countable setscountable sets means that each
Ei has a countable number ofelements.... a sequence means that
the number of Eis is countable.I.e. both.-- Jon Haugsand Dept.
of Informatics, Univ. of Oslo, Norway,
mailto:jonhaug@i.uio.no 
http://www.i.uio.no/~jonhaug/,
Phone: +47 22 85 24 92
===
Subject: Re: countable sets: Let E1,
E2, E3, ... be a sequence of pairwise disjoint countable sets.
Prove: that union of Ek as k = 1 to oo is countable. I do not
need help proving: this. I need someone to explain exactly
what is being asked. In Let E1, E2,: E3,... be a sequence of
pairwise disjoint countable sets is that saying: that the
numbers of Ei's are countable or is it saying that each Ei 
has
a: countable number of elements??Both. The E1, E2, E3, ... part
says there are a countable number of E's,and pairwise 
disjoint
*countable* sets says that each Ei is countable.Ted
===
Subject:
Re: countable sets>Let E1, E2, E3, ... be a sequence of
pairwise disjoint countable sets. Prove>that union of Ek as k
= 1 to oo is countable. I do not need help proving>this. I
need someone to explain exactly what is being asked. In Let
E1, E2,>E3,... be a sequence of pairwise disjoint countable
sets is that saying>that the numbers of Ei's are countable 
or
is it saying that each Ei has a>countable number of
elements??Both. Sequences by denition are countable ordered
sets. In this casethe elements of the sequence are countable
sets.-- I'm not interested in mathematics that might have
anythingto do with reality. -- Russell Easterly, in
sci.math
===
Subject: Re: Need help with proof!> Show that for
arbitrary natural numbers m and n, such that 1<=m<=n the>
following holds: the product n(n-1)(n-2) ...(n-m+2)(n-m+1) is
dividable by> m!Here's a hint that should get you started
without giving away the answer.Show that for every prime that
divides m!, that prime occurs in thefactorization of
n(n-1)(n-2)...(n-m+1) at least as many times as it occursin
the factorization of m!.-- --Tim Smith
===
Subject: Re: Parity
Check Matrix of a Systematic Linear Block Code> The generator
matrix of a systematic linear block code has the> form G = [Ik
: P]. How can it be shown that the parity check> matrix is of
the form H = [-P^T : In-k]?With great ease.If a vector (a : b)
is supposed orthogonal to the code generatedby G then (a :
b)G^t = 0 so a + b P^t = 0 or a = -b P^t etc.Are you in the
IEEE?-- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.htmlLacan,
Jacques, 79, 91-92; mistakes his penis for a square root,
88-9Francis Wheen, _How Mumbo-Jumbo Conquered the
World_
===
Subject: Re: How fast is the Connued Fraction
factorization algorithm? >>I have another question: In each of
the above factorizations, only half>>(approximately) of the
primes in the factor base are used. I read>>somewhere that N
(the composite) must be congruent to a square modulo>>the
primes in the factor base, but I don't see why:>>If x^2 = y
mod N, then x^2 - y = kN, for some k. Now if a prime p
(from>>the factor base) divides y, then we must have p | (x^2
- kN). So I>>conclude that kN must be congruent to a square
modulo p, but that does>>not say anything about N itself?
Please enlighten me. > I don't know anything about the
continued fraction method> (as has been pointed out!)> but if
it is like the simple quadratic sieve> one considers products
of numbers of the form> Q(x) = x^2 - n> (with x close to
sqrt{n})> each of which is smooth.> So one need only consider
small primes p> such that n is a quadratic residue mod p. (In
the multiple polynomial version one takes> Q(x) = ax^2 + 2bx +
c,> with b^2 - ac = n, so that> aQ(x) = (ax + b)^2 - n> and
much the same is true.)Well, using the same notation, then the
continued fraction method deals with the polynomials Q(x) = x^2
- kn.-Michael.
===
Subject: Re: Constraints on Self-Dual Linear
Block Codes> If (n,k) is a q-ary linear block code of length n
and> size q^k, then in order for the code to be self-dual> it
is fairly easy to show that n = 2*k. Are there any other
constraints required of a code> in order for it to be
self-dual?Yes ... it has to satisfy the denition of
self-duality :-)-- Robin Chapman,
www.maths.ex.ac.uk/~rjc/rjc.htmlLacan, Jacques, 79, 91-92;
mistakes his penis for a square root, 88-9Francis Wheen, _How
Mumbo-Jumbo Conquered the World_
===
Subject: Re: Constraints on
sha1:4g8CzI2Px2pOdQBhceYQt3njFfU=Robin Chapman
a q-ary linear block code of length n and>> size q^k, then in
order for the code to be self-dual>> it is fairly easy to show
that n = 2*k.>> Are there any other constraints required of a
code>> in order for it to be self-dual?>> Yes ... it has to
satisfy the denition of self-duality :-)See the text by
Roman. I know the answer to my question is yes - I was looking
for clarication/insight. -- % Randy Yates % Bird, on the
wing,%% Fuquay-Varina, NC % goes ¤oating by%%% 919-577-9882 %
but there's a teardrop in his eye...%%%% <yates@ieee.org> %
.95One Summer Dream', *Face The Music*,
ELOhttp://home.earthlink.net/~yatescr
===
Subject: Re: What is a
number?/What is not a number?> In
said:>I'll offer this denition: A number is an 
element
of a number set. A>number set is either the set of
integers Z or any ring/eld R that>contains Z.> So
you're excluding the Cayley Numbers? Elements of 
nite elds?>
> Yes. It was my intention to eliminate all nite 
elds or
nite> rings for that matter. Good, that is an opinion.
However, you have to exclude more. Polynomials> in one of more
variables with integer coefcients for instance. They> form a
ring and contain Z. Unless you would call them numbers (and>
that is not common practice). I have no idea how you would
exclude them.As the denition stands, they 
couldn't be
excluded. Of course, I haveavoided dealing with the question
of why we even need a label ofnumber in the rst place. If we
had an answer to that question,we'd have a basis to decide 
how
to proceed. We have three ways toproceed:1) revamp the
denition above to remove polynomials,2) accept polynomials 
as
numbers, or3) dene a number set as any mathematical set at
all.In favor of 1) is a measure of psychological comfort, I
suppose. Infavor of 2) is that polynomials mod ideals are
isomorphic tononpolynomial rings or elds, which are already
included as numbersets. In favor of 3) is that the word number
is just a way to avoidsaying the wordy expression element of a
mathematical set with closedbinary operation. And this brings
us back to Cayley numbers.I suppose the question of what
number means in common practice is atthe heart of this thread.
I know of no commonly accepted denition ofnumber in
mathematics, so any denition is a contender. Thatcertain
mathematical objects are not usually considered as numbersis
of no signicance as far as I'm concerned, 
because I canfind
noparticular rationale to call any object a number.We could
just demand that whatever objects were considered as numbersup
to 1985, say, are grandfathered in as numbers, but no more
forever.Or we could try to come up with a rationale for the
label number andthen craft a denition on top of it.One
rationale for the label number could be to regard all
objectsused directly for counting or measuring as numbers.
That includesthe naturals, the integers, the rationals, and
the reals, but nothinglarger than that. And that leaves some
sets out which are already usedas number sets, such as the
complex numbers. But I'm not fond of thisrationale. However, 
I
can't think of any rationale to dene 
numberwhich won't either
leave some sets out or include too many.Patrick
===
Subject: Re:
What is a number?/What is not a number?> In
said:>I'll offer this denition: A number is an 
element
of a number set. A>number set is either the set of
integers Z or any ring/eld R that>contains Z.> So
you're excluding the Cayley Numbers? Elements of 
nite elds?>
> Yes. It was my intention to eliminate all nite 
elds or
nite> rings for that matter. Good, that is an opinion.
However, you have to exclude more. Polynomials> in one of more
variables with integer coefcients for instance. They> form a
ring and contain Z. Unless you would call them numbers (and>
that is not common practice). I have no idea how you would
exclude them.As the denition stands, they 
couldn't be
excluded. Of course, I haveavoided dealing with the question
of why we even need a label ofnumber in the rst place. If we
had an answer to that question,we'd have a basis to decide 
how
to proceed. We have three ways toproceed:1) revamp the
denition above to remove polynomials,2) accept polynomials 
as
numbers, or3) dene a number set as any mathematical set at
all.In favor of 1) is a measure of psychological comfort, I
suppose. Infavor of 2) is that polynomials mod ideals are
isomorphic tononpolynomial rings or elds, which are already
included as numbersets. In favor of 3) is that the word number
is just a way to avoidsaying the wordy expression element of a
mathematical set with closedbinary operation. And this brings
us back to Cayley numbers.I suppose the question of what
number means in common practice is atthe heart of this thread.
I know of no commonly accepted denition ofnumber in
mathematics, so any denition is a contender. Thatcertain
mathematical objects are not usually considered as numbersis
of no signicance as far as I'm concerned, 
because I canfind
noparticular rationale to call any object a number.We could
just demand that whatever objects were considered as numbersup
to 1985, say, are grandfathered in as numbers, but no more
forever.Or we could try to come up with a rationale for the
label number andthen craft a denition on top of it.One
rationale for the label number could be to regard all
objectsused directly for counting or measuring as numbers.
That includesthe naturals, the integers, the rationals, and
the reals, but nothinglarger than that. And that leaves some
sets out which are already usedas number sets, such as the
complex numbers. But I'm not fond of thisrationale. However, 
I
can't think of any rationale to dene 
numberwhich won't either
leave some sets out or include too many.Patrick
===
Subject: Re:
What is a number?/What is not a number?> In
said:>I'll offer this denition: A number is an 
element
of a number set. A>number set is either the set of
integers Z or any ring/eld R that>contains Z.> So
you're excluding the Cayley Numbers? Elements of 
nite elds?>
> Yes. It was my intention to eliminate all nite 
elds or
nite> rings for that matter. Good, that is an opinion.
However, you have to exclude more. Polynomials> in one of more
variables with integer coefcients for instance. They> form a
ring and contain Z. Unless you would call them numbers (and>
that is not common practice). I have no idea how you would
exclude them.As the denition stands, they 
couldn't be
excluded. Of course, I haveavoided dealing with the question
of why we even need a label ofnumber in the rst place. If we
had an answer to that question,we'd have a basis to decide 
how
to proceed. We have three ways toproceed:1) revamp the
denition above to remove polynomials,2) accept polynomials 
as
numbers, or3) dene a number set as any mathematical set at
all.In favor of 1) is a measure of psychological comfort, I
suppose. Infavor of 2) is that polynomials mod ideals are
isomorphic tononpolynomial rings or elds, which are already
included as numbersets. In favor of 3) is that the word number
is just a way to avoidsaying the wordy expression element of a
mathematical set with closedbinary operation. And this brings
us back to Cayley numbers.I suppose the question of what
number means in common practice is atthe heart of this thread.
I know of no commonly accepted denition ofnumber in
mathematics, so any denition is a contender. Thatcertain
mathematical objects are not usually considered as numbersis
of no signicance as far as I'm concerned, 
because I canfind
noparticular rationale to call any object a number.We could
just demand that whatever objects were considered as numbersup
to 1985, say, are grandfathered in as numbers, but no more
forever.Or we could try to come up with a rationale for the
label number andthen craft a denition on top of it.One
rationale for the label number could be to regard all
objectsused directly for counting or measuring as numbers.
That includesthe naturals, the integers, the rationals, and
the reals, but nothinglarger than that. And that leaves some
sets out which are already usedas number sets, such as the
complex numbers. But I'm not fond of thisrationale. However, 
I
can't think of any rationale to dene 
numberwhich won't either
leave some sets out or include too many.Patrick
===
Subject: Re:
What is a number?/What is not a number?>I'll offer this
denition: A number is an element of a number set. A>number
set is either the set of integers Z or any ring/eld R
that>contains Z.>So ordinal numbers are not numbers?-- Stephen
J. Herschkorn herschko@rutcor.rutgers.edu
===
Subject: Re: What
is a number?/What is not a number?>>>I'll offer this
denition: A number is an element of a number set. A>number
set is either the set of integers Z or any ring/eld R that>
>contains Z.> So ordinal numbers are not numbers?Good point.
OK, a number is an element of a number set. A number setis
eithera) the set of integers Z orb) any ring/eld/algebra R
that contains Z orc) any of the ordinal or cardinal objects
dened by Cantor.I think it's beginning to look 
like a weaker
denition is in orderfor number set. Let's look 
at some
possibilities. A number set is1) any set that has a closed
binary operation dened on all pairs of elements of the 
set,2)
any semigroup, or3) any totally orderable set.But these
possible denitions may be too weak. The advantage of 1)
isthat it allows people to call the elements of their favorite
setnumbers if they choose to do so. It seems to me that pure
whim isthe driving force behind what set elements have been
generallyaccepted to be called numbers.Maybe the desire to
label set elements as numbers lies in thepsychological need of
people to reify their favorite sets. And if thishypothesis of
ontology being at the root of the number issue istrue, then
the term imaginary number is an oxymoron, or at the veryleast
an equivocation. Perhaps the orgin of the number concept
isPlatonistic at heart. In any case, it seems too late to
return to thetime when number was associated with counting or
measuring. Then whatis a number?Patrick
===
Subject: Re: What is
a number?/What is not a number?>>I'll offer this 
denition:
A number is an element of a number set. A>>number set is either
the set of integers Z or any ring/eld R that>>contains Z.>
So ordinal numbers are not numbers?It seems even the natural
numbers are not numbers.-- Dave SeamanJudge Yohn's mistakes
revealed in Mumia Abu-Jamal
ruling.<http://www.commoncouragepress.com/index.cfm?action=
book&bookid=228>
===
Subject: Re: What is a number?/What is not
a number?>>I'll offer this denition: A number 
is an
element of a number set. A>>number set is either the set of
integers Z or any ring/eld R that>>contains Z.> So
ordinal numbers are not numbers? It seems even the natural
numbers are not numbers.I addressed that issue on my post of
the fourteenth.Patrick
===
Subject: Re: What is a number?/What
is not a number?@mozo.cc.purdue.edu:>I'll offer this
denition: A number is an element of a number set. A>number
set is either the set of integers Z or any ring/eld R
that>contains Z.>> So ordinal numbers are not numbers? It
seems even the natural numbers are not numbers. > So Numbers
are not numbers;)-- If its Monday then I am a fool but not
ignorant.Its Wenesday 
===
Subject: Re: What is a number?/What
is not a number?> @mozo.cc.purdue.edu: >>>I'll offer
this denition: A number is an element of a number set. A>
>number set is either the set of integers Z or any ring/eld
R that>contains Z.>>> So ordinal numbers are not
numbers?> It seems even the natural numbers are not
numbers.> So Numbers are not numbers> ;)The question is,
What is it we really mean by the label number?There are two
ways to dene the set of number sets: One way is tosimply 
list
all number sets, the other is to give a rule (given by
adenition) that takes the place of the listing of number
sets. Myquestion is, Is there a denition that provides a 
rule
that createsthe same set of number sets as there are in the set
of number setscreated as a listing by at?So, we have the set
of number sets S = {N, Z, Rationals, Reals,Complex, Cayley,
...}. So, what denition of number gives us theset S?Look at
it this way: Let S' = {M : P(M), where M is a
mathematicalset}. What denition do we propose such that 1) M
is a number set iffP(M) is true, and 2) such that S' = S?Is
there a mathematical or set-theoretic property P(M) common to
allthose number sets M listed in S (other than convention says
thatthey should be listed there)?Patrick
===
Subject: Matrix
convert from 1 coordinate space to another where they-axis is
inverted.In the 1st space, the 3x3 transformation matrix is
specied as with +ve yupwards:a b 0c d 0e f 1in the 2nd space
:a c eb d f0 0 1where in the 2nd space y is downwards.I've
worked out by trial and error that by negating b & c the
conversion issuccessful so thata -c e-b d f0 0 1will make
everything work for my conversion.what I understand is to do a
¤ip of y-axis one needs to multiply the matrixby :1 0 00 -1 00
0 1but that doesn't givea -c e-b d f0 0 1Is there somehting
more to a transpose than simply ¤ipping along
===
Subject: Re: A trend ?> Hi Dan
Weiner,> Re: How I feel about your one word posts,> You
added two words: duly noted ,> Do I detect a trend here ?
But will his next post have three words or four?Reminds me of
a story told about President Calvin Coolidge, known asSilent
Cal.A woman reporter once approached him and said, My editor
bet me thatI couldn't get more than two words out of you.The
President replied, You Lose.Double-A
===
Subject: x^2 + y^4 =
z^4+ y^4 = z^4 has no positive-integer solutions. Is the proof
of this result short enough for some kind soul to post it, or
need I make a trip to the library? (I have citations.)--
Stephen J. Herschkorn herschko@rutcor.rutgers.edu
===
Subject:
Re: x^2 + y^4 = z^4
===
Subject: x^2 + y^4 = z^4 >x^2 + y^4 =
z^4 has no positive-integer solutions. Is the proof of >this
result short enough for some kind soul to post it, or need I
>make a trip to the library? (I have citations.)When youfind
proof, let us know.It's one that's been 
pestering me for
ages.Here's my archived thoughts upon x^2 + y^4 = z^4 for
pairwise coprime x,y,zIf y odd: y^4 = z^4 - x^2 = (z^2 -
x)(z^2 + x); factors odd, coprime y^4 = u^4 v^4; x = (u^4 -
v^4)/2; z^2 = (u^4 + v^4)/2If x odd: x^2 = z^4 - y^4 = (z -
y)(z + y)(z^2 + y^2); factors odd, coprime x^2 = u^2 v^2 w^2;
y = (u^2 - v^2)/2; z = (u^2 + v^2)/2 w^2 = z^2 + y^2 = (u^4 +
v^4)/2Thus problem boils down to showing no coprime a,b,c with
a^4 + b^4 = 2c^2when a /= b.Is there some coprime a,b,c with
a^2 + b^2 = 2c^2and a /= b?-- this is innite descent proof,
not quickno solution in positive integers to the equation x^4
+ y^4 =
z^2http://www.math.toronto.edu/mathnet/plain/questionCorner/
fermat4.html----
===
Subject: Re: x^2 + y^4 = z^4> Subject: x^2
+ y^4 = z^4> Thus problem boils down to showing no coprime
a,b,c with> a^4 + b^4 = 2c^2> when a /= b. Is there some
coprime a,b,c with> a^2 + b^2 = 2c^2> and a /= b?Yes, 1^2 +
7^2 = 2*5^2 7^2 + 23^2 = 2*17^2 , etca^2 + b^2 = 2c^2suggests
you are looking at the normof (1+i)(p +iq)^2 in Z(i)This
givesa = p^2 - q^2 - 2pqb = p^2 - q^2 + 2pq(a,b) = 1, (p,q)=1
=> p,q not both oddSome kind of descent might be possible in
===
Zor even more likely in Z(i)Subject: Re: x^2 + y^4 =
z^4
===
Subject: x^2 + y^4 = z^4>> Thus problem boils down to
showing no coprime a,b,c with> a^4 + b^4 = 2c^2> when a /=
b.>> Is there some coprime a,b,c with> a^2 + b^2 = 2c^2> and a
/=
b?>********************************************************
Well, (2m^2-1)^2 + (2m^2-4m+1)^2 =
2(2m^2-2m+1)^2._______________________________________________
__________Eric J. Wingler (wingler@math.ysu.edu)Dept. of
Mathematics and StatisticsYoungstown State UniversityOne
University PlazaYoungstown, OH
44555-0001330-941-1817
===
Subject: Re: x^2 + y^4 = z^4>Is there
some coprime a,b,c with> a^2 + b^2 = 2c^2>and a /= b?There are
innitely many positive integer solutions to, for example,1^2
+ b^2 = 2*c^2 (see Pell's equation).Robert Israel
israel@math.ubc.caDepartment of Mathematics
http://www.math.ubc.ca/~israel University of British Columbia
Vancouver, BC, Canada V6T 1Z2
===
Subject: tensors for totsHow
far can one get in understanding the structure of tensors at
apoint on a manifold in terms of our old school friends column
vectors,row vectors and matrices? Even in general relativity
all we have ateach point on the manifold is a real vector
space with fourcomponents, so it seems we can go all the way
-- but I'm havingtrouble with some of the mappings.Shall I
show what work you've done so far, or would somebody care
totake the ball from there... ?
===
Subject: Re: tensors for
tots> How far can one get in understanding the structure of
tensors at a> point on a manifold in terms of our old school
friends column vectors,> row vectors and matrices? Even in
general relativity all we have at> each point on the manifold
is a real vector space with four> components, so it seems we
can go all the way -- but I'm having> trouble with some of 
the
mappings. Shall I show what work you've done so far, or 
would
somebody care to> take the ball from there... ? You can get
very far. Since you can't even understand the word 
structure,
never tensors, or even such nonsense as curved space-time
without vectors. First you build a 3 dimensional school. Then
you build an 4-dimensional box. Then you build an
n-dimensional table. Then you rotate the table n+1 times, so
that 1 is now at the origin. Then you build a radioactive GPS
robot orbiting the Earth. Then you tell mathematicians to go
back to school, and teach chemists something about logic,
since not only doesn't cold-fusion work at negative
temperatures, it doesn't even work at imaginary
temperatures.
===
Subject: Re: tensors for tots> How far can one
get in understanding the structure of tensors at a> point on a
manifold in terms of our old school friends column vectors,>
row vectors and matrices?Very far, I think. Start with a
vector space. A vector is clearlyindependent of his
coordinates referring to a special basis, so howdo the
coordinates change, when we change the basis? The answer
ofLinear Algebra is: Let B be the matrix of the change of the
basis,and v be the (column-)vector of coordinates with respect
to the old basis. Then the new coordinates w are given byw =
B^(-1) v.B^(-1)'s name in SRT is often Lambda mu nu.A vector
is a contravariant vector, so to say, an (1,0)-Tensor.Now
consider a linear mapping from the vector space to the eld
ofscalars, represented by a row v*. How do the coordinates
change? LAsays, the new coordinates w* arew* = v*
B^T(Multiplication from the right with the Transposed of
B)B^T's name in SRT is often Lambda tilde (swung dash) mu 
nu.A
co-vector is a covariant vector, an (0,1)-Tensor.A Matrix
consists of rows and colums, so it is a (1,1)-Tensor.Generally
speaking, a (m,n)-Tensor is a tuple of m contravariantvectors
(= vectors, tangent vectors) and n covariant vectors(=
co-vectors, cotangent vectors).OK?Best wishes, Alm
===
Subject:
Re: tensors for tots> How far can one get in understanding
the structure of tensors at a> point on a manifold in terms
of our old school friends column vectors,> row vectors and
matrices? Very far, I think. Start with a vector space. A
vector is clearly> independent of his coordinates referring to
a special basis, so how> do the coordinates change, when we
change the basis? The answer of> Linear Algebra is: Let B be
the matrix of the change of the basis,> and v be the
(column-)vector of coordinates with respect to the old >
basis. Then the new coordinates w are given by> w = B^(-1) v.>
B^(-1)'s name in SRT is often Lambda mu nu.> A vector is a
contravariant vector, so to say, an (1,0)-Tensor.> Now
consider a linear mapping from the vector space to the eld
of> scalars, represented by a row v*. How do the coordinates
change? LA> says, the new coordinates w* are> w* = v* B^T>
(Multiplication from the right with the Transposed of B)>
B^T's name in SRT is often Lambda tilde (swung dash) mu nu.> 
A
co-vector is a covariant vector, an (0,1)-Tensor.> A Matrix
consists of rows and colums, so it is a (1,1)-Tensor.>
Generally speaking, a (m,n)-Tensor is a tuple of m
contravariant> vectors (= vectors, tangent vectors) and n
covariant vectors> (= co-vectors, cotangent vectors).> OK?Darn
... I've been given a certain number of credits here,
butsquandered a number of them by loosely parsing the question
-- like agenie who grants three wishes, and debits one when you
say I wish Ihad more time to think.I'm familiar with the 
idea
of change of basis and of matrices actingon vectors to express
the vectors in terms of new bases, but I'm notentirely sure 
why
you call the inverse of (the thing which multipliescolumn
vectors to get new forms of column vectors) the matrix of
thechange of basis. I would just call the thing itself by this
title: v' = Bv (column vectors, square matrix). But let that
pass.I am also familiar with the idea of transforming matrices
themselvesto a new basis. If B is the transformation matrix (in
my sense),then the prescription for transforming a matrix A is:
A' = BAB^(-1)which can be read to say that, given a vector
expressed in the newbasis, we rst transform it back to the
old basis, operate with theold matrix, then transform again to
the new basis. (You apparentlywould reverse the symbols B and
B^(-1)). I'm sure you can followso far what 
I'm exrpessing on
my simple minded level, even if youmight prefer to express it
slightly differently.Now here is where I run into difculty:
if we say that v' = Bv (mynotation), then I'm 
uncertain how we
wish to transform v*(v-transpose, a row vector). Or for that
matter, say we want v* tobe the dual vector corresponding to
v: I'm uncertain when we wouldsimply want to set v* = v^t, 
and
when we want to do something morecomplicated.I have a hunch
this has to do with the metric tensor ... oh yeah, wehaven't
introduced that yet. I also have a hunch this has to do
withwhether we content ourselves with orthogonal
transformations ofcoordinates, so that B^1 = B^t, or go for
the gusto with general(invertable) linear transformations.This
is the level I would like to be able to understand with
dinkymatrices before trying to understand the differential
manifold case,since as you seem to agree the
somethingomorphism is complete at eachpoint. I also hope not
to get hung up on any terminological problemsbetween math and
physics, perhaps by saying things like we'd likethis object 
to
be invariant, instead of using terms like covariantand
contravariant, to start.Anyway, to take one more baby step, if
I have not exhausted mycredits, am I correct in thinking that a
reasonable way to introduce ametric tensor into babyland would
be to dene the inner product ofa row vector and a column
vector, in a particular coordinate system,to be a bilinear
form with a matrix sandwiched in the middle, thatmatrix our
metric?
===
Subject: No Set Contains Every Computable Natural
(was Church-Turing compared to Zuse-Fredkin thesis)Stephen
Dave Seaman:> A set is recursive if there is an effective
procedure (a Turing machine,> perhaps) that can decide
membership in the set. That is, the machine> halts and returns
1 if the input is in the set, and the TM halts and> returns 0
if the input is not in the set.This is the denition of a
decidable language at
wikipedia:http://en.wikipedia.org/wiki/Decidable_languageA
decidable or recursive language is a formal language that is a
recursiveset, i.e., for which there exists an algorithm to
solve the followingdecision problem: Given string w, does w
belong to the language? Thealgorithm is not allowed to run
into an innite loop and has to produce aYES/NO answer for 
any
input string after a nite amount of time. Toformalize the
rather vague term algorithm, one usually employs
Turingmachines, but several other equivalent approaches are
possible.This denition should say nite number 
of steps
instead of niteamount of timeI will come back to this.This 
is
the denition of a recursively enumerable
language:http://en.wikipedia.org/wiki/Recursively_enumerable_
languageDenition 1. Given string w as input, the algorithm
halts and outputs YESif and only if w belongs to the language
L. If w does not belong to thelanguage L, the algorithm either
runs forever, or halts and outputs NO.There is a second
denition that you might want to read.Let me 
dene a language,
L, that consists of all unary representations ofnatural
numbers.1 = one, 11 = two, 111 = three, etc.This language is
recursively enumerable. Given a string, x, there exists aTM
thatwill halt after a nite number of steps if x is a member
of L.This language is NOT decidable. Consider what happens if
x is an innitestringof 1's. The TM will not 
halt after a
nite number of steps.The set of all natural numbers is not a
recursive set.Consider what happens when we talk about the set
of all TM's.Assume we can encode the state transition table 
of
a TM usinga natural number. This set can not be recursive for
the samereason that the natural numbers are not recursive.
Given astring, i, we can not be sure that TM(i) is actually a
TMbecause we can not be sure that i is actually a natural
number.No set can contain every computable natural
number.Proof:Assume a TM produces a tape with all of the
natural numbersencoded in unary where each natural is followed
by a 0 (or a blank).Turing gives the state transition table of
such a TM 010110111011110...I can dene a three state TM that
willfind a natural numberthat is not on this tape:1) Read
right until there is a 0.2) Read right until a second 0 is
found.3) Backup and write a 1 on the previous 0.Repeat steps
(1) through (3).This TM will always produce a tape that has
exactly one 0.This 0 will be at a nite position on the tape
and the string of 1'sthat preceds this 0 represents a 
natural
number that was not onthe original input tape.The Halting
Problem is ill posed.There is no set that contains every TM
just as there is no setthat contains every computable natural
number.Now, about the difference between nite amount of time
vsnite number of steps. There are theories of computing
wherea TM can perform an innite number of steps in 
nite
time.These are called
hypercomputers.http://en.wikipedia.org/wiki/HypercomputerA
non-deterministic Turing machine is an example of a
hypercomputer.Another example is an accelerated Turing
Machine. An ATM performsthe rst operation in one unit of
time, the second operation is 1/2 unitof time, the third
operation in 1/4 unit of time, etc. An ATM will performan
innite number of operations after two units of time.Assume 
we
have a computer that can perform an operation in onenanosecond.
The language, L, that I dened above is not evenrecognizable 
if
we expect an answer before the universe freezes over.We can
easily compute a natural number, n, that is so large thatour
nanosecond computer will take 15 billion years to recognizen
as a natural number. There is only a nite number of
naturalsless than or equal to n. There is an innite number 
of
naturalslarger than n. Nearly all natural numbers are larger
than n.So, we see that we have to assume that a TM can
performan innite number of operations in nite 
time to even
saythat L is recursively enumerable.Russell- Zeno was right.
Motion is impossible.
===
Subject: Re: No Set Contains Every
Computable Natural (was Church-Turing compared to Zuse-Fredkin
of all natural numbersis the function (lambda x )[1]. (I am
using Church's excellent lambdanotation for functions.) All
constant functions are recursive. Fora trivial proof see:
_Computability and Unsolvability_ by Martin Davisor _Theory of
Recursive Functions and Effective Computability_ by Rogers.In
fact, in the latter source, see page 6 where the author 
denes
theclass of primitive recursive functions. This is quite a
standard denition(although equivalent alternatives exist in
the literature). You willsee that (lambda x)[1] is primitive
recursive by denition. It is well-knownthat the primitive
recursive functions are recursive (and indeed thisis often
part of the denition of recursive function). Therefore,
theset of natural numbers is not only recursive, it is
primitive recursive.This is all quite trivial and should be
mastered on the rst day of studyingrecursion theory.-Leonard
Blackburn
===
Subject: Re: No Set Contains Every Computable
function of the set of all natural numbers> is the function
(lambda x )[1]. (I am using Church's excellent lambda>
notation for functions.) All constant functions are recursive.
For> a trivial proof see: _Computability and Unsolvability_ by
Martin Davis> or _Theory of Recursive Functions and Effective
Computability_ by Rogers.> In fact, in the latter source, see
page 6 where the author denes the> class of primitive
recursive functions. This is quite a standarddenition>
(although equivalent alternatives exist in the literature).
You will> see that (lambda x)[1] is primitive recursive by
denition. It iswell-known> that the primitive recursive
functions are recursive (and indeed this> is often part of the
denition of recursive function). Therefore, the> set of
natural numbers is not only recursive, it is primitive
recursive.> This is all quite trivial and should be mastered
on the rst day ofstudying> recursion theory.>I 
haven't read
Computability and Unsolvability orTheory of Recursive
Functions and Effective ComputabilityI didfind a reference to
Theory of Recursive Functions and EffectiveComputability at
Mathworld:http://mathworld.wolfram.com/
GeneralRecursiveFunction.htmlThere are two camps of thought on
the meaning of general recursive function.One camp considers
general recursive functions to be equivalent to the
usualrecursive functions. For members of this camp, the word
general emphasizesthat the class of functions includes all of
the specic subclasses, such asthe primitive recursive
functions (Rogers 1987, p. 27).The other camp considers
general recursive functions to be equivalent tototal recursive
functions.This is Mathwrld's denition of total 
recursive
functions:http://mathworld.wolfram.com/RecursiveFunction.htmlA
recursive function is a function generated by (1) addition,
(2)multiplication, (3) selection of an element from a list,
and (4)determination of the truth or falsity of the inequality
a < b according tothe technical rules:1. If F and the sequence
of functions G1,G2,G3, ..., Gn are recursive, thenso is
F(G1,G2,...Gn).2. If F is a recursive function such that there
is an x for each a with ,then the smallest x can be obtained
recursively.A Turing machine is capable of computing recursive
functions.My approach assumes computable means TM computable.My
argument falls into the other camp.I am using the denition 
of
decidable language given at wikipedia:A decidable or recursive
language is a formal language that is a recursiveset, i.e., for
which there exists an algorithm to solve the followingdecision
problem: Given string w, does w belong to the language?
Thealgorithm is not allowed to run into an innite loop and
has to produce aYES/NO answer for any input string after a
nite amount of time. Toformalize the rather vague term
algorithm, one usually employs Turingmachines, but several
other equivalent approaches are possible.All regular,
context-free and context-sensitive languages are recursive,
butthere exist recursively enumerable languages that are not
recursive; oneexample is given by the halting problem.I have
shown why a Turing machine can not decide if a string, w,
representsa natural number.wikipedia says several other
equivalent approaches are possible.Roger's 
denition can not
be one of these equivalent approaches.The set of all natural
numbers is not a recursive set if our denitionof recursive
set requires there exist a TM that can decide membershipin the
set. Mathworld's denition indicates there are
differencesbetween denitions of general recursive
function.This appears to be one of those differences.Russell-
2 many 2 count
===
Subject: Re: No Set Contains Every Computable
Natural <XqCdnXyGSYcqprDdRVn-gw@comcast.com>
<YRcXb.2828$tL3.1683@newsread1.news.pas.earthlink.net>
<if6dnWHskv8BmbPdRVn-uw@comcast.com>
<O1xXb.12743$B53.5007@newssvr29.news.prodigy.com>
<R--dndv_wuIXILPdRVn-uA@comcast.com>
<HqCXb.25527$Co6.14111@newssvr25.news.prodigy.com>
<keSdnWEKapvdv63dRVn-uQ@comcast.com>
<LhXXb.13180$gP1.7891@newssvr29.news.prodigy.com>
<odydnXiTFf4ZQqzdRVn-ig@comcast.com>
<aPudnWROE-hbLq_dRVn-tw@comcast.com>> I am using the 
denition
of decidable language given at wikipedia: A decidable or
recursive language is a formal language that is a recursive>
set, i.e., for which there exists an algorithm to solve the
following> decision problem: Given string w, does w belong to
the language? The> algorithm is not allowed to run into an
innite loop and has to produce a> YES/NO answer for any 
input
string after a nite amount of time. To> formalize the rather
vague term algorithm, one usually employs Turing> machines,
but several other equivalent approaches are possible. Note
that it says a nite amount of time.> All regular,
context-free and context-sensitive languages are recursive,>
but there exist recursively enumerable languages that are not
recursive;> one example is given by the halting problem. Did
you realize that the natural numbers in unary (or binary, or
any standard base) is a *regular* language, and hence
context-free, and recursive (i.e. decidable.). You reference
this site, and then you try to make claims contradicting
them.> I have shown why a Turing machine can not decide if a
string, w,> represents a natural number. That's because you
think the string of an innite number of 1s is a natural
number. Once again, you have to understand that every natural
number is nite, even in unary representation.> The set of 
all
natural numbers is not a recursive set if our denition> of
recursive set requires there exist a TM that can decide
membership> in the set. Mathworld's denition 
indicates there
are differences> between denitions of general recursive
function.> This appears to be one of those differences. Did
you even notice that your denitions of these sets are based
on *formal languages* ? Did you even look up what a formal
language is? Here you go, from Wikipedia:> In mathematics,
logic and computer science, a formal language is a> set of
nite-length words (or strings) over some nite 
alphabet. Note
that a formal language is a set of FINITE-LENGTHED
strings.J
===
Subject: Re: No Set Contains Every Computable
Natural (was Church-Turing compared to Zuse-Fredkin
message>> Stephen Harris <stephen.p.harris@sbcglobal.net>
if there is an effective procedure (a Turing machine,>
perhaps) that can decide membership in the set. That is, the
machine> halts and returns 1 if the input is in the set, and
the TM halts and> returns 0 if the input is not in the
set.> This is the denition of a decidable language at
wikipedia:> http://en.wikipedia.org/wiki/Decidable_language>>
A decidable or recursive language is a formal language that is
a recursive> set, i.e., for which there exists an algorithm to
solve the following> decision problem: Given string w, does w
belong to the language? The> algorithm is not allowed to run
into an innite loop and has to produce a> YES/NO answer for
any input string after a nite amount of time. To> formalize
the rather vague term algorithm, one usually employs Turing>
machines, but several other equivalent approaches are
possible.>> This denition should say nite 
number of steps
instead of nite> amount of time> I will come back to this.>A
TM is an abstract logical computing machine (Turing: LCM)which
does not consume physical resources. Those Super-Turingmachines
you mention later have an abstract property that TMsdo not.
They represent the real numbers (...Pi) with innite
precision.This is not physically realizable.> This is the
denition of a recursively enumerable language:>
http://en.wikipedia.org/wiki/Recursively_enumerable_language>>
Denition 1. Given string w as input, the algorithm halts and
outputs YES> if and only if w belongs to the language L. If w
does not belong to the> language L, the algorithm either runs
forever, or halts and outputs NO.>> There is a second
denition that you might want to read.>> Let me 
dene a
language, L, that consists of all unary representations of>
natural numbers.> 1 = one, 11 = two, 111 = three, etc.>> This
language is recursively enumerable. Given a string, x, there
exists a> TM that> will halt after a nite number of steps if
x is a member of L.> This language is NOT decidable. Consider
what happens if x is an innite> string> of 
1's. The TM will
not halt after a nite number of steps.>> The set of all
natural numbers is not a recursive
set.>http://www.cs.auckland.ac.nz/CDMTCS/chaitin/
omega.htmlAnyway, it was Post who put his nger on the
essential idea, which is thenotion of a set of objects that
can be generated one by one by a machine, insome order, any
order. In other words, there is an algorithm, a
computerprogram, for doing this. And that's the essential
content of the notion of aFAS, that's the toy model that 
I'll
use to study the limits of the formalaxiomatic method. I 
don't
care about the details, all that matters to me isthat 
there's
an algorithm for generating all the theorems, that's the
keything.And since this is such an important notion, it would
be nice to give it aname! It used to be called an r.e. or
recursively enumerable set. Theexperts seem to have switched
to calling it a computably enumerable orc.e. set. I'm 
tempted
to call it something that Post said in one of hispapers, which
is that it's a generated set, one that can be generated byan
algorithm, item by item, one by one, in some order or other,
slowly butsurely. But I'll resist the temptation!FAS = c.e.
set of mathematical assertionsSo what's the bottom line? 
Well,
it's this: Hilbert's goal of one FAS forall of 
math was an
impossible goal, because you can't put all ofmathematical
truth into just one FAS. Math can't be static, 
it's got to
bedynamic, it's got to evolve. You've got to 
keep extending
your FAS, addingnew principles, new ideas, new axioms, just as
if you were a physicist,without proving them, because they
work! Well, not exactly the way thingsare done in physics, but
more in that spirit. And this means that the ideaof absolute
certainty in mathematics becomes untenable. Math and physics
maybe different, but they're not that different, not as
different as peoplemight think. Neither of them gives you
absolute certainty!SH: I brought this up because
of:...According to Turing's denition, a number 
is computable
its decimal expansion can be written down by a circle
freemachine. This is a machine that performs an
nitely many,Turing denes the machine to be 
circular.
Circular machinesreach a conguration from which there is no
possible move,or go on moving, but do not print any more
symbols.SH: There is a signicant difference between a
circle-freemachine that works at printing out the innitely
long digitsof Pi and a circular machine that may move but not
printany more output --> no symbols so it is no longer
generating.A generated set, one that can be generated by an
algorithm,item by item, one by one, in some order or other,
slowly but surely.Computing Pi is not an example of the
.95halting problem' justbecause it does not halt. 
The circular
machine is an exampleof the halting problem. I keep getting
the impression that youthink these are analgous situations
just because they dont halt.COMPUTABILITY AND RECURSION 285
Robert SoareDenition 1.2. (i) A function is Turing 
computable
if it is denable bya Turing machine, as dened 
by Turing 1936.
(See [Kleene, 1952] or[Soare, 1987].)(ii) A set A is computably
enumerable (c.e.) if A is (/) or is the rangeof a Turing
computable function.(iii) A function f is recursive if it is
general recursive, as dened byG ~odel 1934. (See also
Kleene's variant 1936, 1943, and [1952, p. 274].)(iv) A set 
A
is recursively enumerable (r.e.) if A is (/) or is the range
ofageneral recursive function.Recursively Enumerable Reals and
Chaitin Omega NumbersCristian S. Calude, Peter H. Hertling,
Bakhadyr Khoussainov, Y. WangA real number alpha is called
recursively enumerable if it is the limit ofarecursive,
increasing, converging sequence of rationals.Roughly speaking,
the computational depth of an object is the amount of
timerequired for an algorithm to derive the object from its
shortestdescription. Bennett showed that the characteristic
sequence XK of thehalting problem is strongly deep, while no
random sequence is strongly deep.Investigating this matter
further, Juedes, Lathrop, and Lutz have consideredthe notion
of usefulness'' of innite 
sequences. A sequencex is useful if
all recursive sequences can be computed with oracle accessto x
within a xed recursive time bound. For example XK is useful,
whileno recursive or random sequence is useful.> Consider what
happens when we talk about the set of all TM's.> Assume we 
can
encode the state transition table of a TM using> a natural
number. This set can not be recursive for the sameWikipedia:A
set S of natural numbers or tuples of natural numbers, or of
literalstrings, is recursively enumerable or computably
enumerable orsemi-decidable if it satises either (and
therefore both) of the followingequivalent conditions. a..
There is an algorithm that, when given a natural number n (or
tuple ofnatural numbers, or word, as the case may be)
eventually halts if n is amember of S and otherwise runs
forever. a.. There is an algorithm that generates the members
of S. That meansthat its output is simply a list of the
members of S: s1, s2, s3, ... Ifnecessary it runs
forever.Natural numbers have two main purposes: they can be
used for counting(there are 3 apples on the table), or they
can be used for ordering(this is the 3rd largest city in the
state).> reason that the natural numbers are not recursive.
Given a> string, i, we can not be sure that TM(i) is actually
a TM> because we can not be sure that i is actually a natural
number.>Some problems are recursively enumerable but not
recursive. Oneexample is the halting problem, which is the
problem: Given a program and input parameters, will that
program eventually halt when run with those parameters?> No
set can contain every computable natural number.>> Proof:>
Assume a TM produces a tape with all of the natural numbers>
encoded in unary where each natural is followed by a 0 (or a
blank).> Turing gives the state transition table of such a TM>
010110111011110...>> I can dene a three state TM that will 
nd
a natural number> that is not on this tape:>> 1) Read right
until there is a 0.> 2) Read right until a second 0 is found.>
3) Backup and write a 1 on the previous 0.> Repeat steps (1)
through (3).>> This TM will always produce a tape that has
exactly one 0.> This 0 will be at a nite position on the 
tape
and the string of 1's> that preceds this 0 represents a 
natural
number that was not on> the original input tape.>SH: The
successor function builds the whole set of natural
numbersWikpedia:a.. There is a natural number 0.a.. Every
natural number a has a successor, denoted by a + 1.a.. There
is no natural number whose successor is 0.Distinct natural
numbers have distinct successors: if a ? b, then a + 1 ? b+1If
a property is possessed by 0 and also by the successor of every
naturalnumber which possesses it, then it is possessed by all
natural numbers.(This postulate ensures that the proof
technique of mathematical inductionis valid.)> The Halting
Problem is ill posed.> There is no set that contains every TM
just as there is no set> that contains every computable
natural number.>The set of natural numbers in countably
innite and has no greatestor last digit. The set can be
represented as {0,1,2,3,...}which doesnot list the entire set
or assume it has a largest digit.just as there is no set that
contains every computable natural numberIf you want to
establish a set that has one-to-one correspondencebetween TMs
which terminate after 1 step, 2 steps, 3 steps and so on,and
the natural numbers, it seems reasonable to me. What
systematicmethod do you have that eliminates the computable
natural numbersand excludes them from the set of real numbers?
I suspect you meanthe computable numbers cannot be sorted out
of the reals. I amnot sure there is an algorithm to list
computable transcendentals.I am not much interested in those
===
Subject: Re:
No Set Contains Every Computable Natural (was Church-Turing
compared to Zuse-Fredkin thesis)Russell Easterly
Dave Seaman:> A set is recursive if there is an effective
procedure (a Turing machine,> perhaps) that can decide
membership in the set. That is, the machine> halts and
returns 1 if the input is in the set, and the TM halts and>
returns 0 if the input is not in the set.> This is the
denition of a decidable language at wikipedia:>
http://en.wikipedia.org/wiki/Decidable_language>> A decidable
or recursive language is a formal language that is a
recursive> set, i.e., for which there exists an algorithm to
solve the following> decision problem: Given string w, does w
belong to the language? The> algorithm is not allowed to run
into an innite loop and has to produce a> YES/NO answer for
any input string after a nite amount of time. To> formalize
the rather vague term algorithm, one usually employs Turing>
machines, but several other equivalent approaches are
possible.>> This denition should say nite 
number of steps
instead of nite> amount of time> I will come back to this.>>
This is the denition of a recursively enumerable language:>
http://en.wikipedia.org/wiki/Recursively_enumerable_language>>
Denition 1. Given string w as input, the algorithm halts and
outputs YES> if and only if w belongs to the language L. If w
does not belong to the> language L, the algorithm either runs
forever, or halts and outputs NO.>> There is a second
denition that you might want to read.>> Let me 
dene a
language, L, that consists of all unary representations of>
natural numbers.> 1 = one, 11 = two, 111 = three, etc.>> This
language is recursively enumerable. Given a string, x, there
exists a> TM that> will halt after a nite number of steps if
x is a member of L.> This language is NOT decidable. Consider
what happens if x is an innite> string> of 
1's. The TM will
not halt after a nite number of steps.>You 
can't input an
innite string to the tape. If you could, it 
wouldn'tbe a
Turing machine.> The set of all natural numbers is not a
recursive set.>No natural number requires an innite unary
representation. All may berepresented using a nite number of
1s.> Consider what happens when we talk about the set of all
TM's.> Assume we can encode the state transition table of a 
TM
using> a natural number. This set can not be recursive for the
same> reason that the natural numbers are not recursive. Given
a> string, i, we can not be sure that TM(i) is actually a TM>
because we can not be sure that i is actually a natural
number.>All Turing machines can be nitely represented. There
is no such thing asa TM transition function that requires an
innite state automata forrepresentation.> No set can contain
every computable natural number.>> Proof:> Assume a TM
produces a tape with all of the natural numbers> encoded in
unary where each natural is followed by a 0 (or a blank).This
is impossible, a) a decider is limited to a nite number of
stepsbefore halting, b) to output all integers in unary
requires an inifnitenumber of steps - hence, no TM can produce
this output.> Turing gives the state transition table of such a
TM> 010110111011110...>> I can dene a three state TM that 
will
nd a natural number> that is not on this tape:>> 1) Read 
right
until there is a 0.> 2) Read right until a second 0 is found.>
3) Backup and write a 1 on the previous 0.> Repeat steps (1)
through (3).>> This TM will always produce a tape that has
exactly one 0.> This 0 will be at a nite position on the 
tape
and the string of 1's> that preceds this 0 represents a 
natural
number that was not on> the original input tape.>You have not
at all proven this. First, how can there be a 0 at a
niteposition from the start? If there is, there must a 0 to
its right (in factan innite number of zeros), hence step 2
will read this second zero, andstep 3 will go back and erase
the .95remaining 0'. The part about the 
numberpreceding the last
0 being an integer not on the tape makes no sense either.> The
Halting Problem is ill posed.> There is no set that contains
every TMSure there is : A = {<M> | M is a valid TM
representation}It is easy to check, given a data structure,
whether the input is a valid TMor not.> just as there is no
set> that contains every computable natural number.>All
natural numbers are computable. However, they _all_ can't be
generatedin a single run of any algorithm/TM.> Now, about the
difference between nite amount of time vs> 
nite number of
steps. There are theories of computing where> a TM can perform
an innite number of steps in nite time.> These 
are called
hypercomputers.> http://en.wikipedia.org/wiki/Hypercomputer>>
A non-deterministic Turing machine is an example of a
hypercomputer.No, an NTM is not a hyper-computer since any NTM
can be simulated by astandard TM. The NTM must be given
additional abilities to becomesuper-Turing.> Another example
is an accelerated Turing Machine. An ATM performs> the rst
operation in one unit of time, the second operation is 1/2
unit> of time, the third operation in 1/4 unit of time, etc.
An ATM will perform> an innite number of operations after 
two
units of time.>> Assume we have a computer that can perform an
operation in one> nanosecond. The language, L, that I dened
above is not even> recognizable if we expect an answer before
the universe freezes over.> We can easily compute a natural
number, n, that is so large that> our nanosecond computer will
take 15 billion years to recognize> n as a natural number.
There is only a nite number of naturals> less than or equal
to n. There is an innite number of naturals> larger than n.
Nearly all natural numbers are larger than n.>> So, we see
that we have to assume that a TM can perform> an innite
number of operations in nite time to even say> that L is
recursively enumerable.>To say that there is some natural
number that a physical computer will neverbe able to print out
in unary is one thing, but to say that a TM (a
purelymathematical object) 
.95can't' print out certain natural
numbers in unary in anite number of steps is not 
correct.l8r,
Mike N. Christoff
===
Subject: Re: No Set Contains Every
Computable Natural (was Church-Turing compared to Zuse-Fredkin
thesis)Michael N. Christoff <mchristoff@sympatico.caREMOVETHIS>
This is the denition of a recursively enumerable language:>>
> This language is recursively enumerable. Given a string, x,
there existsa> TM that> will halt after a nite number of
steps if x is a member of L.> This language is NOT
decidable. Consider what happens if x is an>> innite string
of 1's. The TM will not halt after a nite 
number ofsteps.>
> You can't input an innite string to the 
tape. If you
could, it wouldn't> be a Turing machine.>No, but you can 
input
nite instructions which generate innite 
output.Russell is
right about Turing saying this, and I found it repeated:A
Highly Random Number by Becher, Daicz, and Chaitin:In this
note we present a natural example of a random numberthat goes
beyond the class of omega-numbers. The idea goesback to
Turing's celebrated paper On computable numbers,with an
application to the Entscheidungsproblem [11],where he
describes the computable numbers as the real numberswhose
decimal expansion is calculable by nite means.According to
Turing's denition, a number is computable 
its decimal
expansion can be written down by a circle freemachine. This is
a machine that performs an unendingcomputation in the course of
machine to be
circular. Circular machinesreach a conguration from which
there is no possible move,or go on moving, but do not print
any more symbols.SH: You might not want to call this an
algorithm which hasa terminating clause, but it is a
computable method. See 1) FinitenessKnuth, Vol. 1, Sec.
1.1:The modern meaning for algorithm is quite similar to that
of_recipe_,_process_, _method_, _technique_, _procedure_,
_routine_,_rigmarole_, except that the word algorithm connotes
somethingjust a little different. Besides merely being a 
nite
set ofrules that gives a sequence of operations for solving a
specictype of problems, an algorithm has ve 
important
features: 1) Finitness. An algorithm must always terminate
after a nite number of steps. [...] (A procedure that has 
all
of the characteristics of an algorithm except that it possibly
lacks _nitness_ may be called a _computational method_.
[...])(A procedure which has all the characteristics of an
algorithm except that it possibly lacks nitness may be 
called
acomputational method. Besides his algorithm for the
greatestcommon divisor of two integers, Euclid also gave a
geometrical construction that is essentially equivalent to
algorithm E, except that it is procedure for obtaining the
``greatest common measure'' of the lengths of 
two line
segments; this is a computational method that does not
terminate if the given lengths are 
``incommensurate''). [SH:
for intererested constructivists] 2) Denitess. Each step of
an algorithm must be precisely dened; the actions to be
carried out must be rigorously and unambiguously specied for
each case. [...] 3) Input. An algorithm has zero or more
_inputs_: quantities that are give to it initially before the
algorithm begins, or dynamically as the algorithm runs. [...]
4) Output. An algorithm has zero or more _outputs_: quantities
that have a specied relation to the inputs. [...] 5)
Effectiveness: An algorithm is also generally expected to be
_effective_, in the sense that its operations must all be
sufciently basic that they can in principle be done exactly
and in a nite length of time by someone using pencil and
paper. [...](A procedure which has all the characteristics of
an algorithm except that it possibly lacks nitness may be
called acomputational method. Besides his algorithm for the
greatestcommon divisor of two integers, Euclid also gave a
geometrical construction that is essentially equivalent to
algorithm E, except that it is procedure for obtaining the
``greatest common measure'' of the lengths of 
two line
segments; this is a computational method that does not
terminate if the given lengths are 
``incommensurate'').I think
Church's (1937a) description agrees with 
innite symbols.The
author [i.e. Turing] proposes as a criterion that an
innitesequence of digits 0 and 1 be computable that it shall
bepossible to devise a computing machine, occupying a nite
spaceand with working parts of nite size, which will write
down thesequence to any desired number of terms if allowed to
run for asufciently long time. As a matter of convenience,
certain furtherrestrictions are imposed on the character of
the machine, but theseare of such a nature as obviously to
cause no loss of generality -in particular, a human
calculator, provided with pencil and paperand explicit
instructions, can be regarded as a kind of Turing machine.The
Broad Conception of Computation by Jack CopelandIs addition
over the computable numbers a computablefunction? (That is, is
x+y computable whenever x and yare both computable numbers?)
The answer is .95yes', butoff the top of 
one's head one might
think otherwise,for if x (or y) is a computable number having
an innitedecimal representation, how can x be input, given
Turing'srestriction that the input inscribed on the tape
mustconsist of a nite number of symbols?The solution is to
input x in the form of a program which,if inscribed on the
(otherwise blank) tape of some universalTuring machine, would
cause the machine to calculate thedecimal representation of x
digit by digit. Nothing trickyis going on here. The program
inscribed on the tape is, afterall, a sequence of symbols, and
this particular sequence ofsymbols has no less a claim to be
counted as a representationof the number, x, than 
.9514' and
.951110' have to be counted asrepresentations of 
the number
fourteen.In short, Turing has given us a new method for
representingnumbers; and in this system there are nite
representationsof some of the numbers which, in the decimal
system, can berepresented only by means of an innite 
sequence
of symbols.To return to the machine that is to add x and y,
where x andy are any computable numbers: once the
representations, inTuring's ingenious system, of x and y 
have
been inscribed onthe tape, the machine is set in motion. First
it calculatesthe rst digit of the decimal representation of 
x
andremembers it; next it calculates the rst digit of
therepresentation of y; then it adds these two digits, using
aleft[CapitalEth]to[CapitalEth]right addition procedure; then
it turns to the seconddigits of the representations, and so
on. Each digit of thedecimal representation of x+y is produced
by the machine insome nite number of steps.I hope the
discussion so far has afforded some insight intowhat a
computable number is, in Turing's restricted sense
of'computable number'. A number is computable, 
in his
sense,just in case the number can be expressed by means of a
certainnite string of symbols, namely a string of symbols
which, inscribed on the otherwise blank tape of some
universal Turingmachine, will cause the machine to churn out
the decimalrepresentation of the number.SH: I think this means
that a TM can add Pi and e digit by digit.I posted not to argue
on Russell's side, but because I thought thisTuring 
ingenuity
deserved some recognition. I think the terminatingrequirement
makes more sense when applied to a digital computer.I think
there is one term. not halting, doing double duty for
twodifferent reasons. Printing Pi can continue to output
symbols, acircle-free machine, but a circular machine, can
continue to movebut doesn't output symbols. So it is no 
longer
a .95generative' case.BTW, Chaitin has a new, 
warm, e-book online
about the historyand philosophy of developing primes,
randomness and
Omega.http://www.cs.auckland.ac.nz/CDMTCS/chaitin/
omega.htmlMETA MATH! The Quest for Omega by Gregory
===
Subject: Re: No Set Contains Every
Computable Natural (was Church-Turing compared to Zuse-Fredkin
message>> Michael N. Christoff
message>>> This is the denition of a
recursively enumerable language:>> This language is
recursively enumerable. Given a string, x, thereexists> a>
TM that> will halt after a nite number of steps if x is a
member of L.> This language is NOT decidable. Consider
what happens if x is an>> innite string of 
1's. The TM will
not halt after a nite number of> steps.>>> You 
can't
input an innite string to the tape. If you could, 
itwouldn't>
> be a Turing machine.> No, but you can input nite
instructions which generate innite output.> Russell is right
about Turing saying this, and I found it repeated:>But that
would be a different language. The language would then be {<M>
| Mis a TM that outputs a string a nite string of 1s}. This
language isclearly undecidable. But that does not prove his
point, as any languagewhere (language U lang_complement) is
composed of arbitrary TMs and whosemembership is based on
determining non-trivial properties of those TMs, isundecidable
(a la Rice's theorem). However L is certainly
decidable:<quote>Let me dene a language, L, that consists of
all unary representations ofnatural numbers.1 = one, 11 = two,
111 = three, etc.</quote>l8r, Mike N. Christoff
===
Subject: Re:
No Set Contains Every Computable Natural (was Church-Turing
compared to Zuse-Fredkin thesis)Michael N. Christoff
in> message>> Russell Easterly <logiclab@comcast.net>
>> This is the denition of a recursively enumerable
language:>> This language is recursively
enumerable. Given a string, x, there> exists> a> TM
that> will halt after a nite number of steps if x is a
member of L.> This language is NOT decidable. Consider
what happens if x is an>> innite string of 
1's. The TM
will not halt after a nite numberof> steps.>>>
> You can't input an innite string to the 
tape. If you could,
it> wouldn't be a Turing machine.>>> No, but you can
input nite instructions which generate 
inniteoutput.>
Russell is right about Turing saying this, and I found it
repeated:> But that would be a different language. The
language would then be {<M> |M> is a TM that outputs a string
a nite string of 1s}. This language is> clearly undecidable.
But that does not prove his point, as any language> where
(language U lang_complement) is composed of arbitrary TMs and
whose> membership is based on determining non-trivial
properties of those TMs, is> undecidable (a la Rice's
theorem). However L is certainly decidable:>> <quote>> Let me
dene a language, L, that consists of all unary
representations of> natural numbers.> 1 = one, 11 = two, 111 =
three, etc.> </quote> l8r, Mike N. Christoff> You can't
input an innite string to the tape. If you could, it> 
wouldn't
be a Turing machine.>>Harris:> No, but you can input
nite instructions which generate 
inniteoutput.> Russell is
right about Turing saying this, and I found it repeated:Chaitin
et al: This is a machine that performs an unendingcomputation
symbols.[SH: If you read the thread, Russell previously
claimed this.}Harris added in last post:I posted not to argue
on Russell's side, but because I thought thisTuring 
ingenuity
a TM and you respondedas if I supported Russell's argument 
and
brought up languages,which I fail to see as germane to my
informational post. Mymotive rests in the posts I've read on
Google which have usedthis nite input stipulation to claim
that a TM cannot generatea potentially innite output. But
more to your point, I don't seewhy there can't 
be two inputs,
x and y, a fragment of innite Pi(or any unique generic
fragment) which is then nitely matchedagainst the 
potentially
innite output of a Pi computational method.I think a mutual
sequence match would eventually be discoveredalthough when
this match would occur is undecidable. This is aspecualtive
post on my part and I'm willing to change my
===
Subject: Re: No Set Contains Every
Computable NaturalMichael N. Christoff
dene a language, L, that consists of all unary
representationsof> natural numbers.> 1 = one, 11 = two,
111 = three, etc.>> This language is recursively
enumerable. Given a string, x, there existsa> TM that>
will halt after a nite number of steps if x is a member of
L.> This language is NOT decidable. Consider what happens if
x is aninnite> string> of 1's. The TM will not 
halt after
a nite number of steps.> You can't input an 
innite
string to the tape. If you could, it wouldn't> be a Turing
machine.This is equivalent to saying that a human operator
must decide ifthe tape has an innite string of 
1's before
giving the tape to theTM. There is no automated way to decide
if the tape has aninnite string of 
1's.Requiring the input
tape to be nite is a severe limitation.For example, no
language, L, could contain every binaryrepresentation of
rational numbers between 0 and 1.1/3 has an innite binary
representation in base 2.Every denition of 
TM's that I have
seen assumesthat an innite tape of blanks is allowed.> The
set of all natural numbers is not a recursive set.> No
natural number requires an innite unary representation. All
may be> represented using a nite number of 1s.There is no TM
that can decide if the string has aninnite string of 
1's.>
Consider what happens when we talk about the set of all 
TM's.>
> Assume we can encode the state transition table of a TM
using> a natural number. This set can not be recursive for
the same> reason that the natural numbers are not recursive.
Given a> string, i, we can not be sure that TM(i) is actually
a TM> because we can not be sure that i is actually a natural
number.> All Turing machines can be nitely represented.
There is no such thingas> a TM transition function that
requires an innite state automata for> representation.There
is no TM that can decide if the transition table isinnitely
long.> No set can contain every computable natural number.>
>> Proof:> Assume a TM produces a tape with all of the
natural numbers> encoded in unary where each natural is
followed by a 0 (or a blank).>> This is impossible, a) a
decider is limited to a nite number of steps> before 
halting,
b) to output all integers in unary requires an inifnite> number
of steps - hence, no TM can produce this output.>> Turing
gives the state transition table of such a TM>
010110111011110...Turing's TM doesn't produce 
a unary
representation of every natural?> I can dene a three state
TM that willfind a natural number> that is not on this
tape:>> 1) Read right until there is a 0.> 2) Read right
until a second 0 is found.> 3) Backup and write a 1 on the
previous 0.> Repeat steps (1) through (3).>> This TM
will always produce a tape that has exactly one 0.> This 0
will be at a nite position on the tape and the string of 
1's>
> that preceds this 0 represents a natural number that was not
on> the original input tape.> You have not at all proven
this. First, how can there be a 0 at a nite> position from 
the
start? If there is, there must a 0 to its right (infact> an
innite number of zeros), hence step 2 will read this second
zero, and> step 3 will go back and erase the .95remaining 
0'.
The part about thenumber> preceding the last 0 being an
integer not on the tape makes no senseeither.Is doesn't 
matter
that there are an innite number of 0's on the 
inputtape.Every
0 on the original input tape is in a nite position.The TM I
describe will produce an output tape with exactly one 0.My TM
doesn't ever write a 0, so the 0 was on the input tape.> The
Halting Problem is ill posed.> There is no set that contains
every TM>> Sure there is : A = {<M> | M is a valid TM
representation}It is impossible to decide if M is a valid
representation.> It is easy to check, given a data structure,
whether the input is a validTM> or not.It is impossible for a
TM to decide if M is a valid representation.> just as there
is no set> that contains every computable natural number.>
> All natural numbers are computable. However, they _all_
can't begenerated> in a single run of any algorithm/TM.The
output of a TM can't contain a representation of every
natural?> Now, about the difference between nite amount of
time vs> nite number of steps. There are theories of
computing where> a TM can perform an innite number of steps
in nite time.> These are called hypercomputers.>
http://en.wikipedia.org/wiki/Hypercomputer>> A
non-deterministic Turing machine is an example of a
hypercomputer.>> No, an NTM is not a hyper-computer since any
NTM can be simulated by a> standard TM. The NTM must be given
additional abilities to become> super-Turing.I misread the
examples given at wikipedia. It says:A non-deterministic
Turing machine which has a preference ordering over itsnal
states.The additonal ability is preference ordering over its
nal states.> Another example is an accelerated Turing
Machine. An ATM performs> the rst operation in one unit of
time, the second operation is 1/2unit> of time, the third
operation in 1/4 unit of time, etc. An ATM willperform> an
innite number of operations after two units of time.>>
Assume we have a computer that can perform an operation in
one> nanosecond. The language, L, that I dened above is not
even> recognizable if we expect an answer before the universe
freezes over.> We can easily compute a natural number, n,
that is so large that> our nanosecond computer will take 15
billion years to recognize> n as a natural number. There is
only a nite number of naturals> less than or equal to n.
There is an innite number of naturals> larger than n.
Nearly all natural numbers are larger than n.>> So, we see
that we have to assume that a TM can perform> an innite
number of operations in nite time to even say> that L is
recursively enumerable.> To say that there is some natural
number that a physical computer willnever> be able to print out
in unary is one thing, but to say that a TM (a purely>
mathematical object) .95can't' 
print out certain natural numbers
in unary ina> nite number of steps is not correct.I 
didn't say
that a TM can't print out certain natural numbers.I said that 
a
TM that requires a xed, nite amount of time 
toperform an
operation can't print certain natural numbers in areasonable
amount of time.Russell- The universe is one
dimensional