From newshub.ccs.yorku.ca!ists!helios.physics.utoronto.ca!news-server.csri.toronto.edu!rpi!usc!cs.utexas.edu!uunet!psinntp!cambridge.oracorp.com!ian Thu Jan 16 17:20:16 EST 1992
Article 2699 of comp.ai.philosophy:
Xref: newshub.ccs.yorku.ca comp.ai.philosophy:2699 sci.philosophy.tech:1840 sci.logic:797
Newsgroups: comp.ai.philosophy,sci.philosophy.tech,sci.logic
Path: newshub.ccs.yorku.ca!ists!helios.physics.utoronto.ca!news-server.csri.toronto.edu!rpi!usc!cs.utexas.edu!uunet!psinntp!cambridge.oracorp.com!ian
>From: ian@cambridge.oracorp.com (Ian Sutherland)
Subject: Re: Penrose on Man vs. Machine
Message-ID: <1992Jan14.162419.21604@cambridge.oracorp.com>
Organization: ORA Corp, 675 Mass Ave, Cambridge, MA 02139
References: <1992Jan13.175936.2755@oracorp.com> <1992Jan13.200000.7489@husc3.harvard.edu>
Date: Tue, 14 Jan 92 16:24:19 GMT

In article <1992Jan13.200000.7489@husc3.harvard.edu> zeleny@zariski.harvard.edu (Mikhail Zeleny) writes:
>MZ:
>>> While you are at it, feel free to characterize "sufficiently
>>> well" the predicate "... is finite" in a first-order language of your
>>> choice. I eagerly await the results.
>
>DMC:
>>A set is finite if there is no function mapping it onto a proper
>>subset of itself. That is definable in ZFC, if you interpret "function"
>>as a set of ordered pairs.
>
>Note that you are quantifying over the non-denumerable set of all functions
>from N to N; in a countable submodel of V you would end up recognizing all
>sort of interesting infinite sets as finite in virtue of the fact that the
>requisite bijection would exist in a standard model of ZFC, but won't be
>found in any of its countable submodels.

(1) I don't think talking about models has any relevance to this
discussion at all.  Saying that the ZFC definition of something may
not match the "real concept" because ZFC has no way of "knowing" that
its quantifiers really range over "the real universe of sets" doesn't
distinguish it from human beings.  How do you "know" that your use of
quantifiers really corresponds to quantification over "the real
universe of sets"?  The very question is silly: your use of
quantifiers refers to "the real universe of sets" simply because that
is the semantics you use for your utterances.  If this can be taken
to be true of your utterances, then it can be taken to be true of
theorems of ZFC as well.  Adopting this semantics for ZFC, and
assuming that neither you nor ZFC make any false statements about the
mathematical world, the question then becomes, "can you prove more
than ZFC proves?"  You seem to be claiming that you can.  Daryl is
asking for evidence of this.  I would be interested in your answer to
this question in any case Mr. Zeleny.  Do you believe you can prove
more than ZFC can prove?  If so, can you give me some examples?

(2) I'm rather puzzled by your statement about finiteness not being
absolute for countable submodels.  If you're not talking about
transitive sets whose membership relation is the restriction of the
true membership relation, then of course finiteness may not be
absolute, but so what?  If the set isn't transitive, or has some other
membership relation than true membership, then there's no reason to
expect there to be any connection between an element of the submodel
being a finite set and its representing a finite set in the submodel.
If you ARE talking about transitive sets with the true membership
relation, then I believe finiteness is absolute for submodels of ZFC.

>DMC:
>>>> There is no evidence that humans can go beyond formalized reasoning, so
>>>> it is not clear what you are demanding be explained.
>
>MZ:
>>> On the contrary, any semantic consideration goes beyond formalistic, purely
>>> syntactical symbol manipulation.

Mr. Zeleny, you can formalize some sorts of semantical considerations
within set theory.  What evidence do you have that the semantic
considerations that go on in your head are more powerful than those
that can be formalized in set theory?
-- 
Ian Sutherland                          ian@cambridge.oracorp.com

Sans peur


