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Article 2664 of comp.ai.philosophy:
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>From: daryl@oracorp.com
Subject: Re: Penrose on Man vs. Machine
Message-ID: <1992Jan13.165429.27512@oracorp.com>
Organization: ORA Corporation
Date: Mon, 13 Jan 1992 16:54:29 GMT

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.

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 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. 

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.

Daryl McCullough
ORA Corp.
Ithaca, NY






