Re: THIS STATEMENT HAS NO PROOF IN ANY SYSTEM = true or false?

From: Mike Oliver (mike_lists_at_verizon.net)
Date: 01/25/05 

Date: Tue, 25 Jan 2005 10:11:07 -0600

poopdeville@gmail.com wrote:
> tchow@lsa.umich.edu wrote:
>>Recall 'cid 'ooh's objection to my usage of "true": What does it mean
>>to say that "the cartesian product of nonempty sets is nonempty" is
>>true? My response was, following Tarski, that I was using "true" in an
>>eliminable way, and that by saying "the cartesian product of nonempty
>>sets is nonempty" is true, I was saying no more than that the cartesian
>>product of nonempty sets is nonempty. 'cid 'ooh balked at this,
>>claiming not to understand what it means for the cartesian product of
>>nonempty sets to be nonempty unless I specified a model that it was
>>true *in*.
>
>
> I have no quarrel with Tarski's usage (though the redundancy theory of
> truth has been discredited for years). Your usage was slightly
> different, however. You *asked* if AC was true. Your usage and
> Tarski's are flatly incompatible. To wit -- the answer to the question
> "Is AC true?" is trivially "No" since we can construct sets in ZF for
> which it fails. I very much doubt this is what you want.

"Construct sets in ZF for which it fails"? Hard to figure out
what you mean by this. There are *models* of ZF that *believe*
it fails. Let M be such a model, and let x be an element of M
such that M thinks (i) x does not contain the empty set and (ii)
x has an empty Cartesian product.

Now let x^M (the "M-extension of x") be the set of all y in M
such that M thinks y is an element of x. Then for each y in x^M we
can similarly define y^M, and M is correct that no such y^M is empty. Now
the question is, is there a function that, to each such y^M,
assigns an element of y^M? We know that no such function can
be "in" M--that is, there is no f in M such that, if we write f^M
for the function that sends z^M to f(z)^M, for every z that M
thinks is in the domain of f, then f^M has the required property.
It doesn't follow that no such function exists in reality.

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