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Article 2725 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: <1992Jan14.182546.7560@husc3.harvard.edu>
Date: 14 Jan 92 23:25:43 GMT
Article-I.D.: husc3.1992Jan14.182546.7560
References: <1992Jan13.175936.2755@oracorp.com> <1992Jan13.200000.7489@husc3.harvard.edu> <1992Jan14.162419.21604@cambridge.oracorp.com>
Organization: Dept. of Math, Harvard Univ.
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In article <1992Jan14.162419.21604@cambridge.oracorp.com> 
ian@cambridge.oracorp.com (Ian Sutherland) writes:

>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.

MZ:
>>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.

IS:
>(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?

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" would make no sense whatsoever.
Then again, I never said anything of the sort, being that I am not in the
habit of ascribing propositional attitudes to mathematical theories.  ZFC
neither "knows" its propositions, nor "refers" to "the real universe of
sets"; only the conscious agents who interpret it, can know and refer.  As
for proving more than ZFC can, there certainly is a way: just take ZFC
together with `there exists an uncountable inaccessible cardinal', and you
can prove that ZFC is consistent by exhibiting its model.  The question is
whether the resulting system would also prove every other set-theoretic
proposition, together with its negation; I expect that the answer to this
question will be found as a result of semantical consideration.

IS:
>(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.

My claim is simply that the ZFC definitions of finiteness rely on
quantification over uncountable totalities, and hence are second-order in
their nature, failing in extension in a countable model of ZFC; I can't
believe that so much sound and fury could be produced by mathematically
sophisticated people over this simple textbook point.

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.

IS:
>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?

You can't formalize semantical considerations of set theory within set
theory; being that I believe myself to be capable of such semantical
consideration, I see my reasoning as ipso facto more powerful than that
which can be so formalized.  Again, this point has been made over and over
by far better minds; see page 110 of Penrose's book.  You may disagree, but
kindly do so with an argument, rather than by automatic gainsaying of my
propositions; should you lack such an argument, feel free to appeal to your
intuition of being a Turing machine; however please try to abstain from
soliciting my endless reiteration of the same simple points.

To sum up, my position is that, were I a Turing machine, there would exist
some formal mathematical theory whose meaning I couldn't understand.
Personally, I find this implausible; feel free to judge to the contrary.

>-- 
>Ian Sutherland                          ian@cambridge.oracorp.com
>
>Sans peur


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: Mikhail Zeleny                                                     :
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