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Article 2682 of comp.ai.philosophy:
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>From: zeleny@zariski.harvard.edu (Mikhail Zeleny)
Newsgroups: comp.ai.philosophy,sci.philosophy.tech
Subject: Re: Penrose on Man vs. Machine
Message-ID: <1992Jan13.192559.7488@husc3.harvard.edu>
Date: 14 Jan 92 00:25:58 GMT
References: <1992Jan13.165429.27512@oracorp.com>
Organization: Dept. of Math, Harvard Univ.
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Nntp-Posting-Host: zariski.harvard.edu

In article <1992Jan13.165429.27512@oracorp.com> 
daryl@oracorp.com writes:

>DMC is Daryl McCullough
>MZ is Mikhail Zeleny

DMC:
>>> The question of whether a Turing machine program halts on a
>>> given input is definitely *not* second-order! It is perfectly definable
>>> in first-order Peano arithmetic.

MZ:
>> OK.  Define the *intensional* notion of a program halting on a given input
>> without using the second-order notion of finitude.

DMC:
>Look, Mikhail, the question of a Turing machine halting on an input
>involves only existential quantification over integers. There are no
>quantifications over *sets* of integers, and so, therefore, it is a
>first-order statement, and not a second-order statement. If you insist
>that even *mentioning* integers makes it a second-order statement,
>then by that criterion, any mathematical definition is second-order.

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?

DMC:
>In the usual arithmetic hierarchy used by mathematical logicians, you
>characterize arithmetic statements by the number of alternations of
>quantifiers required when the statement is put into normal form (all
>quantifiers out front). The question of whether a Turing machine will
>ever halt on a particular input is an r.e. relation, which is the
>simplest non-recursive relation. It can be defined using a recursive
>relations and a single, existential quantifier. The question of the
>consistency of a theory is co-r.e.; it can be defined using recursive
>relations and a single, universal quantifier. 

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.

DMC:
>The paradigm example of a relation that is *not* first-order is the
>notion of arithmetic truth: that is, the predicate T(x) which holds if
>and only if x is the Godel code of a true statement of (first-order)
>arithmetic.

At the level in which I am interested, that of the intension, even the
G\"odel predicate Bew(p), which is prima facie expressible in the
first-order PA, corresponds to a second-order *property* of p being
provable.  

Once again: do you understand what I mean when I say that finitude is
irreducibly a second-order concept?

>Daryl McCullough
>ORA Corp.
>Ithaca, NY


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: Mikhail Zeleny                                                     :
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