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

poopdeville_at_gmail.com
Date: 01/19/05 

Date: 19 Jan 2005 03:49:36 -0800

tchow@lsa.umich.edu wrote:
> 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.

Where, ZF or ZFC? Remember, models of ZF + "not AC" exist. Without
this crucial bit of information, the question has no (single -- I see
four) answer.

>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.

I didn't say that ZFC is consistent iff it has models. (Though that's
obviously true). What I did say is that ZFC is true relative to a
fixed structure S iff S is a model for ZFC. It is this relativity I
wish to emphasize, since it captures some of the context sensitivity
inherent in mathematical work. For instance, If S is the dihedral
group, S is not a model for ZFC. A group theorist, while working with
D_8, has no interest in the truth of ZFC. Why postulate truths that
don't matter? One would have to make many objectionable ontological
commitments to support these sorts of Platonic truths.

> 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.

If you "all" feel that you can understand the meaning of the
relationships between different sets *intuitively* and unambiguously,
the truth of AC in the sense you describe above is trivial -- just tell
me if your intuitive notion allows a choice function. Brouwer's
intuitive notion didn't. You must be careful when speaking for all
mathematicians.

>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." <snip>

Certainly -- and in contexts in which it's clear I'm working with ZFC,
the statement "Every vector space has a basis" is just as meaningful.
But stripped of it's context, this second statement loses obvious
meaning. I'd have to explain what I meant, coherently, without
assuming that the meaning is obvious (since it isn't).

> 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.
>

Why is the notion of sets and relations more natural than the notion of
a Galois correspondence? Both occur in the "same place" -- ie, have
the same ontological status -- after all. Historically, the notion of
a group was introduced to capture the behavior of certain sets of
automorphisms of roots of polynomials. In fact, one could easily argue
that the group axioms are *more* in line with the intended model than
ZFC, since they obviously satisfy their historical motivations, whereas
it is demonstrable that different notions of sets and elementness
exist.

'cid 'ooh

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