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

tchow_at_lsa.umich.edu
Date: 01/17/05 

Date: 17 Jan 2005 04:08:19 GMT

In article <1105780027.473885.47200@c13g2000cwb.googlegroups.com>,
 <poopdeville@gmail.com> wrote:
>Paraphrasing, amongst other things, Tim Chow asked if AC and the other
>axioms of ZF are true. Unless he was using a non-model-theoretic use
>of the term "true," the ZFC is just as true as the group axioms, since
>we can exhibit models for both sets of axioms.

I used the term the way mathematicians normally use them, which never causes
problems except when someone suddenly decides to get skeptical and ask,
following Pontius Pilate, "What is truth?" For the present purposes, I
can eliminate all uses of the word "true" by simply replacing the statement
of which I am predicating truth with the statements themselves. This is
cumbersome, but is helpful psychologically for those who aren't practiced
in doing such things themselves. So when I ask if AC is true, I am asking
if the Cartesian product of nonempty sets is nonempty. When I ask if "ZFC
is consistent" is true, I am asking if ZFC is consistent. When I ask if
"`Every vector space has a basis' is provable in ZFC" is true, I am asking
if `Every vector space has a basis' is provable in ZFC.

So if you claim not to understand what I mean when I say that "ZFC is
consistent" is true, even after you understand how to eliminate the word
"true" as I have just demonstrated, then you are really claiming not
to understand what I mean when I say that ZFC is consistent. And of
course, in this example, you *do* understand what I mean, because you
make similar assertions yourself, like "ZFC is consistent if and only if
it has models," which you presumably wouldn't make if you didn't know
what such an assertion meant, and which presupposes that you know what
it means for ZFC to be consistent.

The difference between ZFC and the axioms for group theory is not any
kind of interesting structural difference between the first-order theories
themselves, as you point out. The difference is that ZFC is usually used
in the context of trying to capture certain features of general mathematical
discourse---in particular, statements that we make all the time that we feel
we understand the meaning of unambiguously. Although maybe you have not
articulated it to yourself explicitly, I bet you in fact believe that you're
making a specific, meaningful statement when you say "`Every vector space
has a basis' is provable in ZFC." You do not think, "Gosh, what do I really
mean by such a statement? Do I mean that the statement `"Every vector space
has a basis" is provable in ZFC' is true in some model of some formal
system? Which model of which formal system am I talking about?"

The group-theoretic axioms, however, are not introduced in order to capture
statements that mathematicians assert directly without any sense of
ambiguity. Mathematicians do not go around saying things like, "For every
x, y, and z, (x * y) * z = x * (y * z)" without further explanation about
what x, y, and z are and what * is.

The fact that I can take two different "naturally occurring" phenomena
---general mathematical discourse (statements that mathematicians make with
no sense of ambiguity) and statements that apply in the context of groups---
and use the same tool (first-order logic) to analyze both of them does not
mean that the two naturally occurring phenomena are essentially the same,
any more than the fact that I can look at either the moon or a star through
a telescope makes the moon a star.

For example, you can create models of ZFC in which "ZFC is consistent" is
false. Does this make you wonder, "What does it *mean* for ZFC to be
consistent?" (Recall the definitional equivalence between "ZFC is
consistent" and "`ZFC is consistent' is true" that I'm positing.) It
shouldn't. You've known all along what you *mean* by "ZFC is consistent."
The fact that someone can mimic your statement in the first-order language
of set theory and then construct models of ZFC in which your statement
holds and models in which it doesn't is no reason to shake your belief
that you know what it means for ZFC to be consistent.

Similarly, mathematicians agree that they know what it *means* for every
vector space to have a basis (even if they don't agree that it's true).
Mimicking this statement in some formal system doesn't automatically make
it ambiguous. You might, of course, think that the statement really *is*
ambiguous. But this would be for philosophical reasons and not mathematical
ones, and in particular, independence results do not imply that such
statements are ambiguous. 

-- 
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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