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

poopdeville_at_gmail.com
Date: 01/24/05 

Date: 24 Jan 2005 14:01:11 -0800

tchow@lsa.umich.edu wrote:
> In article <vcbsm4rgvpa.fsf@beta19.sm.ltu.se>,
> Torkel Franzen <torkel@sm.luth.se> wrote:
> >poopdeville@gmail.com writes:
> >> It's true in every model for the satisfiability relation.
> > And what does it mean that it's true in every model for the
> >satisfiability relation?
>
> Exactly. However, I suspect that 'cid 'ooh still doesn't see what
I'm
> driving at. I will try again, although I'm beginning to lose hope.
>
> 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.

> My response was that 'cid 'ooh was guilty of exactly the same sin he
was
> charging me with, namely asserting mathematical statements flat out
without
> saying which model of ZFC they were true *in*. As an example, I gave
the
> statement
>
> (*) ZFC is consistent iff ZFC has models.
>
> I asked, what does this mean? I don't understand what it means until
you
> tell me which model of ZFC it's true *in*. (Of course, I do in fact
know
> what it means, but am asking these questions to illustrate my point
that
> 'cid 'ooh's objections can be turned against him.) 'cid 'ooh's
reaction
> was, essentially, that (*) is true in all models of ZFC.

Essentially, my answer doesn't depend on models of ZFC. It depends on
models of the FO meta-language of ZFC. As you note a bit later, you
can push me into an infinite regress by asking "In what model is there
a model of the satisfiability relation?" This supports my claim that
truth is context dependent and not absolute. I'll admit that I didn't
say "... in model X of the satisfiability relation," but I only omitted
it because I assumed context supported it, as it supports most
mathematical lanugage. We only run into trouble when we divorce an
utterance from the context it which it occurs.

The sentence "the cartesian product of non-empty sets is non-empty,"
stripped of its context, is virtually meaningless. Your mind might
wander to thoughts of ZFC or the real numbers, where it's true. But my
mind might wander to ZF, where it can fail to hold. If we're, say,
working on a problem together, the context is fixed, we know if we're
working in ZF or ZFC, and we can determine a definite truth value.

>
> But this is not an answer at all. If it were an answer, then I could
> respond to the question of why I accept that "the cartesian product
of
> nonempty sets is nonempty" by simply observing that it's true in all
> models of ZFC. That's correct, but is obviously not going to satisfy
> 'cid 'ooh. Any axiom is true in all models of that axiom; this
trivial
> fact nothing about whether to accept it, and of course is not why I
> accept it.
>

It certainly would satisfy me. If ZFC is all we're considering, AC is
obviously true. The question gets more interesting if we consider ZF.
(That is, Zermelo-Fraenkel without choice). Remember -- what I'm
advocating is context sensitivity, not absolute truth. A thing can be
true "here" but not "there."

> 'cid 'ooh has a blind spot; when others make mathematical assertions,
> such as "every vector space has a basis," he claims not to be able to
> understand any mathematical statement as having a specific, definite
> meaning unless some model is explicitly specified that it is true
*in*.
> Yet he himself makes mathematical assertions like (*) and like
>
> (**) Theorems of ZFC, e.g., (*), are true in all models of ZFC
>
> baldly, meaning something specific and definite by them, without
saying
> which model of these statements are true *in*. When pressed, he says
> that they are true in all models, which obviously is insufficient to
> fix a specific, determinate meaning for the statements in question,
and
> provides no basis for accepting or rejecting (*) or (**)
>
> What 'cid 'ooh needs to acknowledge is that he makes lots of
mathematical
> statements flat out without needing to say which model they're "true
in."
> As Torkel Franzen points out, an example is (**); surely it cannot be
the
> case that (**) makes no sense unless one says which model it's true
in,
> or else we get an infinite regress (which model is "(**) is true in
model X"
> true in?).
>

I thought you were trying to push me into the infinite regress
direction. Well, there's already an infinite regress in FOL. In fact,
it's almost the same one: every first order language requires a model
theory, which needs an underlying set theoretic structure, which in
turn requires a first-order language. More-or-less, for each language,
you might say you need a new meta-language to understand or "ground"
it. But each meta-language is a language in itself. By induction,
kaboom. This doesn't stop people from using FOL because once you know
what's going in one language/meta-language pair, you know what's going
on in all of them.

Moreover, this sort of infinite regress is to be *expected* from an
infinite sequence of ordered pairs of the form (What does x_{n-1}
mean?, x_n). Every explanation is a symbol that must be interpreted,
and whose meaning is determined by context. This line of thought would
take us far afield. If interested, I suggest Wittgenstein's Blue Book
or Philosophical Investigations.

'cid 'ooh

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