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Article 2590 of comp.ai.philosophy:
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>From: zeleny@zariski.harvard.edu (Mikhail Zeleny)
Newsgroups: comp.ai.philosophy,sci.philosophy.tech,sci.logic
Subject: Re: Penrose on Man vs. Machine
Message-ID: <1992Jan9.021120.7247@husc3.harvard.edu>
Date: 9 Jan 92 07:11:19 GMT
References: <1992Jan8.160615.23680@oracorp.com> <1992Jan9.002732.29965@news.media.mit.edu>
Organization: Dept. of Math, Harvard Univ.
Lines: 54
Nntp-Posting-Host: zariski.harvard.edu

In article <1992Jan9.002732.29965@news.media.mit.edu> 
minsky@media.mit.edu (Marvin Minsky) writes:

>In article <1992Jan8.160615.23680@oracorp.com> 
>daryl@oracorp.com writes:

>>Mikhail Zeleny writes:

MZ:
>>> Since the notions of a program halting on a given input, or a theory
>>> being consistent are fundamentally second-order, i.e. non-recursive,
>>> our ability to understand them is sufficient evidence of our ability
>>> to perform non-algorithmic tasks.

DMC:
>>I almost let this claim slip by without comment, but it is completely
>>incorrect. 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. Perhaps you meant that it is not a
>>*recursive* notion?

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

MM:
>Woops.  The question of whether a Turing machine halts on a
>*particular* input is in fact recursive.  It either does or it
>doesn't, and in the Kleene forumlation of the theory it is a copnstant
>function which is either always 1 or always 0.

Nice try, but no cigar.  First of all, your reading of a *particular* input
is unwarranted by the text of my statement; secondly, I wasn't referring to
the halting problem as such, but to the very concept of a Turing machine
halting on a given input.  The constant function may be recursive, but that
doesn't mean that the corresponding function *in intension* can be
characterized in a first-order language, or computed recursively.

MM:
>I wonder how much of this thread has made the mistake of thinking that

Any attempts at error detection are always appreciated, but never accepted
on anyone's say-so.

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: Connais pas! Connais pas!                                 think    :
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: Mikhail Zeleny                                                     :
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