Re: THIS STATEMENT HAS NO PROOF IN ANY SYSTEM = true or false?
tchow_at_lsa.umich.edu
Date: 01/26/05
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Date: 26 Jan 2005 19:53:12 GMT
In article <ct8928$ijm$1@ra.nrl.navy.mil>,
Ralph Hartley <hartley@aic.nrl.navy.mil> wrote:
>What does it mean for a model to "think" or "believe"? I can guess, but it
>seems an odd usage to me. Is it common?
M thinks that X if X is true in M. It's not really a formal technical term;
it's just an informal locution that is often helpful when trying to keep
track of what's going on in a complicated situation. It's fairly common,
but only when people are being very informal.
>By V do you mean some *particular* model? I think you do, but have no way
>to tell exactly which one. For example, I don't know if V is a model in
>which CH is true or not.
>
>I suppose "the class of all sets" might be seen as specifying V, but I
>don't see how it will help in a discussion with someone who is skeptical of
>set theory.
I do mean some particular model, namely the class of all sets, and I agree
that I wasn't sure it would help in this discussion. But if you've been
following the rest of this thread, you'll see that (to my surprise) it did.
>> Ralph Hartley <hart...@aic.nrl.navy.mil> wrote:
>>>when mathematicians make unqualified statements, with no other
>>>context, they usually *mean* "In ZFC".
>You said:
>> As a matter of sociological fact, this is definitely false.
>
>The only difference I can see between my interpretation and yours is that,
>you refer to a particular model V.
Well, for starters, there is in fact a big difference between ZFC and V.
ZFC is just one system of axioms for sets. If we discarded it, we
could replace it with some other axioms for set theory, but V would
still be around until we decided to abandon Cantor's paradise because
we didn't like infinite sets any more.
More to the point, though, for many mathematical statements, such as
S = "every differentiable function is continuous," it's not even clear
that mathematicians really mean "S is true in V" when they assert S.
Mathematicians feel that they understand what differentiable functions
and continuity are, and although there is a small amount of set theory
buried in the basic definitions, the full glory of V is not necessarily
implicitly present. For example, people have shown that a lot of
analysis can be developed with reference just to integers and sets of
integers (any maybe sets of sets of integers). So when they assert S,
they might only be thinking of S as interpreted in this smaller universe.
There are a lot of options, and unless being precise about the details
of your mathematical universe is important (usually it isn't), people
usually leave these unspecified. The simplest way, I think, of describing
this situation is that mathematicians mean S when they say S. If you
press the question, "But in what universe?" the answer will be "any
suitable universe, sufficient for what I'm studying right now."
This isn't true for all mathematical statements. In particular, for
something like AC, I think that when people assert AC, then they mean
that AC is true in V, because AC is inherently a sweeping set-theoretical
statement about all sets.
>Do different mathematicians mean different things by V, so that some mean a
>model in which CH is true, and some a model in which it is false? Such
>ambiguity would be harmless when discussing statements that don't depend on
>CH etc.
>
>Or is it your position that there is a single "intended" model V and if
>people disagree about its properties, some are right and some are wrong?
The latter.
>Does it make any practical difference which of those positions we accept? I
>seriously doubt it.
Well, I'm not sure how you'd flesh out the concept that "different
mathematicians mean different things by V." So I can't tell if it would
make a practical difference.
-- Tim Chow tchow-at-alum-dot-mit-dot-edu The range of our projectiles---even ... the artillery---however great, will never exceed four of those miles of which as many thousand separate us from the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences
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