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

tchow_at_lsa.umich.edu
Date: 01/27/05 

Date: 27 Jan 2005 16:42:01 GMT

In article <1106823984.872929.137690@z14g2000cwz.googlegroups.com>,
 <poopdeville@gmail.com> wrote:
>Well, you claimed that V was a model for ZFC when you wrote:
>"When I ask whether AC is true, I'm asking whether it's
>true in V, the class of all sets, which is a [proper class] model of
>ZFC."

My bad. Since I do in fact believe that AC is true, I easily slip into
saying things that presuppose its truth. But you're right that if I'm
posing the question "Is AC true?" in what's supposed to be a meaningful
way, then what I said is garbled. If I had been more careful, I would
just have said:

  When I ask whether AC is true, I'm asking whether it's
  true in V, the class of all sets.

I shouldn't have said that V is a model of ZFC in this context, even
though that is commonly assumed in mathematics.

>Explanations are very often successful in
>communicating what we intend. The analogy here is that an explanation
>provides a foundation for what it attempts to explain, just as each
>term in my (possibly transfinite) sequence of models provides a
>foundation for the previous one. Once you get what I'm driving at,
>there's no need for any more.

What I find curious about your account here is that I largely agree with it,
but draw different conclusions than you do from it. You don't think the
regress is infinite because in practice it bottoms out somewhere. I agree,
and describe the situation by saying that at the bottoming-out point, I
grasp your meaning *without* needing to ask for truth-in-a-model. I grasp
your meaning simpliciter. I don't see why you don't describe the situation
the same way.

So for example, I assert AC. You ask, "AC in which model?" I take this to
be a symptom of the fact that you are skeptical about sets, so you can't
grasp AC simpliciter. I've bottomed out at AC; you're bottoming out
somewhere else. But either way, at some point we know what's meant without
having to say "In what model?" So there's some other kind of notion of
truth/meaning coming into play there.

>You don't need to know what numbers are to do number theory. Or sets
>for set theory. Frege struggled for years trying to prove that Julius
>Caesar wasn't a number. But this wasn't for any mathematical reason,
>just a philosophical one. I'm skeptical of your (and my) knowledge of
>what a number is, but I don't claim we can't reason about them.

But don't you at least need to know what symbols are, and what syntactic
rules are? Are you skeptical about rules? I suppose if you like
Wittgenstein, maybe you are. But then how can you do any mathematics
if you don't know what symbols, strings, and rules are?

>Because if you can't communicate your arbitrary meaning of "ZFC is
>consistent," the phrase really is meaningless.

I agree with that. But in practice, there is no trouble communicating
this arbitrary meaning except to extreme skeptics. And we shouldn't
expect to be able to communicate with extreme skeptics.

>Whereas what can be
>proven (and is thus true in all relevant models) is meaningful.

Is it? You have to be able to communicate your proof. What if you can't
do that? USENET provides spectacularly good examples of how even the most
perspicuously transparent proofs fail to be accepted by everyone.

>But what makes you so sure *you* know what a set is if you
>can't even communicate it?

If indeed I couldn't communicate it, I would be worried. But I don't have
any trouble in practice communicating it. I experience no communication
difficulties when I talk with professional set theorists, logicians, and
mathematicians. It's only when talking with students (and on USENET!)
that communication difficulties arise, and there the difficulties arise
not because of lack of clarity of the notion of a set, but for other
reasons.

>I still feel that this division is artificial. A set of axioms for FOL
>mimics the discourse of a particular field.

The set of axioms for a group does not mimic the *discourse* of group
theorists. Group theorists will assert things like, "Every odd order
group is solvable." Or "Every finite simple group is either a cyclic
group of prime order, an alternating group, a group of Lie type, or one
of 26 sporadic exceptions." Or "There exist infinite groups that are
finitely generated with finite exponent." None of these statements can
be expressed in the first-order language of group theory. Only certain
properties of groups---first-order properties---can be expressed in
this language. The function of the group axioms is to define what a
group *is*, and not to mimic *discourse* about groups. 

-- 
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
Loading