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Article 2738 of comp.ai.philosophy:
Newsgroups: comp.ai.philosophy
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>From: daryl@oracorp.com
Subject: Re: Penrose on Man vs. Machine
Message-ID: <1992Jan15.140324.27354@oracorp.com>
Organization: ORA Corporation
Date: Wed, 15 Jan 1992 14:03:24 GMT

Mikhail Zeleny writes:

> In view of non-categoricity of first-order PA, any mathematical
> definition that involves quantification over the integers, whether in
> object language or in meta-language (e.g. by appealing to the
> conventional notion of a proof as a *finite* sequence of propositions)
> is ipso facto second-order.  Is this so hard to understand?

It is hard to understand why you want to distinguish first-order and
second-order definitions if every definition is second-order. Anyway,
I think your use of "second-order" is nonstandard; as I said, the
usual definition has to with whether the definition involves explicit
quantification over sets.

> Thanks for the elementary recursion theory lesson; now that you managed to
> get the didactic compulsion out of your system (I hope), please observe the
> term `intensional' in the above request. Now kindly make an effort to
> discuss the same issue as I am addressing, viz. the question of
> proof-theoretic strength of the language needed to give an exhaustive,
> categorical definition of the above notions.

Mikhail, the fact on which you seem to be basing all this stuff about
the necessity of second-order logic is the following: To uniquely
characterize the natural numbers, one must augment the Peano axioms
with the second-order statement:

     For all sets S of natural numbers, if S contains 0 and is closed
     under the successor operation (if x is in S, then x+1 is in S),
     then S contains all of the natural numbers.

This statement is inherently second-order because there is no way in
first-order logic that one can say "For all sets S". However, in
first-order logic, you can certainly make sure that the above holds
for every *first-order definable* set S. My belief is that that is the
best that we can do in characterizing the naturals, or at least I don't
know of any evidence to the contrary.

Daryl McCullough
ORA Corp.
Ithaca, NY








