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

tchow_at_lsa.umich.edu
Date: 01/28/05 

Date: 28 Jan 2005 16:37:47 GMT

In article <1106877661.601587.62140@f14g2000cwb.googlegroups.com>,
 <poopdeville@gmail.com> wrote:
>Now that we have a common language to work with, many of my claims are
>uncontroversial. However, we've both brought some philosophical
>baggage to the table. In my understanding of your position, you're
>committed to the existence of sets.

To make the main point that truth-in-a-model isn't, and can't be, the
only notion of truth, I don't need to be committed to the existence
of sets.

For me to make statements such as "the cartesian product of nonempty
sets is nonempty" and claim that I'm saying something meaningful, I
do indeed need to be committed to the existence of sets in some sense.
But I don't have to be committed to any kind of platonism. More to
the point, I'm not really committed to the existence of sets any more
than *you* are when you say, "ZFC is consistent iff ZFC has models"
(and think that you're saying something meaningful), because a model is
a set.

>Yes! We need to know what symbols are, and what (the relevant)
>syntactic rules are to do number theory. But the syntactic rules
>demonstrably don't pick out a unique interpretation, which is exactly
>why we need to be careful when talking about truth.

To respond to this let me skip forward a bit to:

>If you can tell me what a set is, outside of a "collection" (the
>more-or-less personal "mental picture") or "an element of a collection
>which satisfies the membership relation" (the public notion that
>doesn't pick out one of these mental pictures), that'd be great. :-)

I'll tell you what a set is if you can tell me what a symbol is.

By the way, historically, symbols were often thought of as being less well
understood than integers. The whole notion of Goedel numbering is, in a
sense, a symptom of the feeling that we know how to deal with integers, so
to convince ourselves that we know how to deal with symbols and strings,
let's convert symbols and strings to integers, so that we *really* know
what we're talking about.

But that's an aside. Let me take your argument about unique interpretations
and turn it against you. Logic, like any other branch of mathematics, can
be formalized and studied. In particular, we can develop a first-order
theory of syntax, which captures discourse about symbols and strings and
so forth. When we do this, we discover that sentences about symbols,
strings, rules, and so forth admit nonstandard interpretations. We might
*think* we know what we mean by a proof of finite length, but lo and behold,
nonstandard models of axioms for syntax show us that "finite length" proofs
could be "nonstandardly finite," which is what most people would call
infinite. Does this shake your belief that you know what symbols and proofs
are?

>> The set of axioms for a group does not mimic the *discourse* of group
>> theorists.
[...]
>I had not thought of this distinction.

Let me elaborate a bit because I think this exercise can be helpful.
Suppose we try to express "every finite simple group is X" in the
first-order language of groups. (I'm being lazy and not saying what
X is, but as you'll see, we won't need to know anything about X other
than that it's some property that groups might have.) As a first step
towards formalization we might try to rephrase:

  For all groups G, if G is finite and G is simple, then G is X.

The problem is that we're quantifying over *groups* here ("for all
groups G"), and in the first-order language of groups we can only
quantify over *group elements*. (That's what "first-order" *means*.)

Let's not give up just yet, though. If we could find a sentence
"FiniteSimple" in the first-order language of groups such that
"GroupAxioms & FiniteSimple" is satisfied by all and only the finite
simple groups, then

   GroupAxioms & FiniteSimple -> X

would be a tolerably good formalization. The problem, though, is
the basic theorem of mathematical logic that any first-order sentence
that is satisfiable by arbitrarily large finite structures is also
satisfiable by some infinite structures. So we can't write down a
sentence that captures "finite" in the sense we want.

Problems also arise with "simple" because this means that there are no
nontrivial normal subgroups. A subgroup is a *subset* of the group,
and again we can't talk directly about subsets in a first-order
language, only about group elements.

These limitations illustrate that when people study the first-order
language of groups, they're *not* trying to mimic a large fraction of
group-theoretic discourse; they're interested specifically in *first-order
properties of groups*, which is a very limited fraction of the properties
of groups that group theorists in general are interested in.

In contrast, when people study ZFC, they are often investigating the
foundations of mathematics as a whole, and it's important that most of
mathematical discourse can be mimicked more-or-less directly in the
language. It's not that we're interested in---for lack of a better
term---"universes" and we're trying to isolate the first-order properties
of universes for special scrutiny. We're actually trying to find formal
counterparts for all the kinds of statements that mathematicians make.

The two projects are quite different and the fact that the same tool
(first-order logic) is being used in both cases does not mean that
direct parallels between ZFC and the axioms for a group always make
sense. 

-- 
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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