mm-441
Subject: Re: need help in underding Torkel's ZFC comment>>
Charlie-Boo says...>> >>1. Would ZFC be better if it included
its rules of inference?For mathematical purposes, the theory
*is* its set of theorems. How>> you describe those theorems as
axioms/axiom schemas/rules of inference>> doesn't really matter
much.>But if different authors use different rules of
inference, the>theorems (and thus the theory) may vary. No.
You really haven't been paying attention:The theorems are the
logical consequences of the axioms.There are various different
ways to formalize the notion oflogical consequence, wch have
different inference rules,but wch are all _equivalent_ - if A
is a logical consequenceof B in one (sound and complete)
formal proof system thenit is also a logical consequence of B
in any other (soundand complete) formal proof system.It's
reable how many times one needs to explain thesetngs to you.
(I mean it's reable, given your assertionthat you underd logic
so much better than the restof us - if you didn't make cracks
like that people mightbe a little more patient with their
explanations...)>And if some are later>discovered to be
inconsistent - oops. Why reinvent the wheel? Those>are just
some of the problems caused by using an incomplete system
and>each author having to fill in the gaps.>> The dard view is
to try to separate>> the rules of *logic*, wch are the same for
any first-order theory,>> from the *axioms*, wch are different
for different theories. The>> advantage is that you can prove
facts about all first-order theories>> (such as compactness)
and then they automatically apply to ZFC, or>> to PA, or to
GNB, etc.>Yes, if the rules of inference are given. But
Zermelo et. al. never>gave rules of inference (although
unintentionally including some>misconstrued as axioms.)For the
fourth time: Ts is because ZFC is a mathematical theory,not a
formal proof system. Mathematical theories do not
containinference rules.>> But for most purposes, it doesn't
really matter how you describe the>> theory.>There's also the
question of efficiency - minimizing the number of>axioms
(dropping redundant ones). Wch of the dard axioms for ZFC
follows from the others?>There are plenty of ramifications>to
using a poorly designed system!>************************

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Subject: Re: need help in underding Torkel's ZFC
commentYou're not interested in whether there are shortcomings
or possible> improvements to a system that you use so much? I
am indeed interested in possible improvements of the public>
transport system, but not in any active way.Do you specialize
in ZFC more than you specialize in use of the
publictransportation system? Do you see a big difference? Or
do you tnkthat's a fair analogy?(To quantify: # of people as
involved in ZFC as you vs. # of peopleinvolved in public
transportation as you. Approximately the same?)> I'm doing
that right now. OK, so I take it that the news postings are a
form of relaxation.Relaxation? Well yeah, they are relaxing,
actually. Don't you tnkposting to news groups is
relaxing?
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Subject: Re: Fast integer division> Is there a
method that allows one to quickly compute integer division
and> remainder of very huge numbers using FFT methods?Someone
else has answered, but for future reference, when asking a
questionlike ts, it is a good idea to specify what you mean by
very huge andwhat kind of hardware you will be using. Many
numbers that were huge enoughto require tngs like FFT methods
a few years ago are now, due to themacnes now being so much
faster, small enough that more straightforwardmethods that
would have been too slow back then might now be faster,
becauseof the overhead of the FFT methods.-- 
===
 Subject: Re:
urgent help neededcan anyone please send me the answer to the
following question. soon> as possible> Let A be a set of
integer. Define an equivalence relation R on AxA.> Show that
the relation you defined is an equivalence relation. Draw a>
diagram for your relation.>>Why is it needed so soon? Your
homework can't possibly be due until>>Monday at the
earliest.>>Doug>On the other hand, you have to sympatze with m
for wanting help, because>the wording of the problem is very
strange. At first, I thought that>the first part of the
question was asking for a definition of>`equivalence
realtion', but after reading the next bit, I presume>it is
supposed to mean Given an example of an equivalence
relation>on AxA. But are you intended to give an example for
all possible sets A, or>can you choose your own A?If the
problem is exactly as he quoted it then it is indeed very
strangely phrased. But it happens a lot around here that
theperson posting a homework problem gets the statement
totallygarbled (presumably because he doesn't underd
themeaning of the words he's garbling) so it's hard to
knowwhere the difficulty really comes
from.>************************