Re: question about axiomatic set theory

Dani wrote:
|NSA involves a similar kind of circularity, that I tried to describe
in
|my last posting - considering that the nonstandard model of N, for
|example, is an enlargement of N, it is a construction that belongs
|properly to logic.

I'm not sure what you mean by "belongs properly to logic".

|But ultraproducts are used in this construction -
|the validity of the axioms of the enlargement is concluded from the
|compactness theorem, a version which uses the axiom of choice in its
|proof. Suppose we don't want the axiom of choice in the basic,
informal
|set "theory" that establishes logic. In that case, there is the same
|"circularity involved in using sets when defining how logic works".

It seems to me that you're using "logic" to refer to two
different kinds of things here, one which set theory depends
on, the other which depends on set theory.

A given formal development can be accompanied by different
informal aspects. A purely formal development is as described
by Hilbert in his Hilbert program: a game played with symbols
according to given rules. The problem with this comes when
one decides one wants the formulas in the game actually to
mean something. But sometimes it's just the formal aspect
that one wants to focus on, and in these cases it's just a
matter of stating what rules one is playing by at any given
time. Understanding the rules requires understanding some
very basic concepts like that of a finite sequence of symbols,
and some simple relationships like one sequence being a
subsequence of another. This means having enough of a
conceptual framework for doing very elementary number theory,
since the natural numbers are just lengths of strings.

In the case of the kind of nonstandard analysis you're
describing, the rules are those of first-order logic plus
the axioms of ZFC (presumably). The illusion of circularity
you've described amounts to the fact that some of the moves
of this game look a lot like discussions of the rules of
*another* game, and how to interpret that other game (an
axiom system for nonstandard analysis). But when it says
things like, "the following axioms have a model...", that
corresponds on a formal level to a somewhat elaborate
formula in the language of ZFC, and the proofs of such claims
correspond to somewhat elaborate derivations in ZFC.

Usually even people adhering to a formalist philosophy of
mathematics don't leave the story just at that. The story
up to this point doesn't offer any justification for the
idea that the results of such a game can be applied to the
real world for instance.

Probably my main point in all this is that there's no way
of conjuring meaning out of the formal aspect after the
fact. You need to have an intended meaning for the formulas
being manipulated that stands independent of the game played.
And the only way that the game serves as a valid way of
generating meaningful truths is if one has a justification
of the rules of the game in terms of that meaning.

The usual interpretation of first-order logic is not too
controversial. It requires that one has a notion of a
"domain" and relations on the domain, plus a standard menu
of connectives, quantifiers, and rules of reasoning.

It's not too hard to imagine someone having such a stash of
basic concepts acquired informally, but to be not so sure
about the *truth* of set theory. I think we may have at
least one newsgroup participant who is this way. Such a
person could look at a proof in ZFC, and instead of merely
looking at it as a game played by certain rules, they could
look at it as a series of inferences about some
(hypothetical) domain of objects.

Such a person needs to distinguish between the notion of
"domain" in the sense needed to explain first-order logic
and the hypothetical objects being discussed while making
deductions from the axioms of ZFC. The latter are being
considered just as the elements of a hypothetical domain
that happens to satisfy the axioms of ZFC.

In order to apply the results of deductions in ZFC to
something else, one needs to justify the truth of the
axioms in a more substantial way. This is what a Platonist
claims to have done. According to a Platonist, who believes
the axioms of ZFC, there's a particular domain of objects
(the "universe of sets") that we have reason to believe
satisfies the axioms of ZFC, and for which the deductive
rules of predicate calculus are valid. If this view is
correct, then applications are correct because they follow
from truth-preserving deductions starting from truths.

|As I noted before, as a way out of this, you might say: ok, now we've
|got logic from basic set theory; now we'll develop ZFC, and then apply
|it back to logic, by (necessarily) considering logic as a
set-theoretic
|(not logical) structure, sort of like a category whose elements are in
|the form (semmantic structure M, formal system F, correspondence
|M-->L(F)). In that context, nonstandard "models" can be defined.

It seems to me that "considering logic as a set-theoretical
structure" could mean a few different things. It could mean
that one has temporarily adopted a convention whereby one
translates statements about logic into set theoretic
language, which is then understood as just being about a
hypothetical structure satisfying the axioms of ZFC. But
something more is required to justify believing that the
translation is equivalent to a more fundamental
interpretation of the statements like we presumably had
before we started doing set theory.

|The problem is that this seems very an undesirable, messy solution -
if
|N* is a set-theoretic structure, it's not really an extension of the
|same N as the one discussed two paragraphs ago.

Suppose we start out by saying N consists of lengths of
finite strings of symbols. It's true that it's a further
step to conclude that the N of set theory is isomorphic to
that. But don't give up so easily on justifying that step.

|This is not a problem in a lot of mathematics, but is only very
|apparent when the object of mathematical inquiry involves mathematics
|itself. In other words, mathematical logic is like philosophizing from
|within the world and language of mathematics (the number 7 woke up one
|day and started to ask "what am I?"), it is self-referring,
|self-conscious so to speak. And on that line, to apply it's methods to
|mathematics is analogous to applying philosophy to politics.

I'm not so sure I agree that the essential problem is
localized to mathematical logic this way. I think the
general problem is how to justify the applicability of
mathematics. Whether it's being applied down to "foundational"
items like symbol strings or "the outside world" the problem
is essentially the same.

Keith Ramsay

.

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