From: zeleny@brauer.harvard.edu (Mikhail Zeleny)
Newsgroups: sci.philosophy.tech,sci.logic
Subject: Re: Sets in mathematics
Date: 14 Apr 92 21:02:04 GMT
Article-I.D.: husc3.1992Apr14.170206.11088
Organization: Dept. of Math, Harvard Univ.

In article <1992Apr12.201239.13391@cambridge.oracorp.com> 
ian@cambridge.oracorp.com (Ian Sutherland) writes:

>In article <s6elfINNjd1@agate.berkeley.edu> 
>epfaith@twinkies.berkeley.edu (Edward Paul Faith) writes:

EPF:
>>Godel's theorems prove that within a certain calculus one could not construct a
>>formal proof of certain facts about that calculus.

IS:
>Godel's theorems apply to a large class of calculi, not just one.
>Perhaps you just misspoke.

On the contrary, perhaps *you* just misspoke, Ian.  Still, many people
share your utterly misguided belief in the mathematical identity of G\"odel
and Rosser.

EPF:
>>Hilbert wanted to formalize
>>all of mathematics into a single calculus (so I recall), and Godel showed that
>>this was impossible, since no matter what calculus we develop, if it is
>>sufficiently complex, we will be able to prove things about it, without being
>>able to carry out the proofs within that calculus.  However, these proofs, while
>>not realizable within the calculus, are nevertheless mathematical proofs. This
>>is why mathematics cannot be completely formalized.

IS:
>This is not a correct interpretation of Godel's theorems.  It is true
>if we apply Godel's theorem to true theories in the language of
>arithmetic.  It is not true, for example, if we apply Godel's theorem
>to ZFC.  Godel's second incompleteness theorem applied to ZFC tells
>us that if ZFC is consistent, then CON(ZFC) is independent of ZFC.
>There is, however, no mathematical proof of the consistency of ZFC.

There is a mathematical proof of the consistency of ZFC, in exactly the
same sense that there is a mathematical proof of the consistency of the
Peano Arithmetic (e.g. by transfinite induction to \epsilon_0).  Example:

ZFC + "there exists a strongly inaccessible cardinal" proves Con(ZFC).

Lest you start quibbling over the status of the strongly inaccessible
cardinal axiom, I submit another example:

ZFC + "there exists a measurable cardinal" proves Con(ZFC + "there exists a
strongly inaccessible cardinal").

I could go on, but I won't.

IS:
>[Aside: There are those who read this newsgroup who believe that we
>can reliably believe CON(ZFC) because of some kind of "inductive"
>reasoning (where "inductive" does not mean "mathematical induction",
>but a nondeductive form of reasoning).  While this may in some sense
>be reasonable, I claim that such "inductive" arguments are not what
>are ordinarily referred to as "mathematical proofs", and that such
>arguments will never appear in mathematical journals as "proofs" that
>CON(ZFC) is a theorem.]

On the contrary, such arguments are often among the most prized results in
mathematics; for example, see Tarski's original paper on inaccessible
cardinals in the 1938 "Fund. Math.".  To those who remain unconvinced by
Ian's oracular pronouncements, I heartily recommend Georg Polya's classic
treatise, "Mathematics and Plausible Reasoning".

EPF:
>>My careless wording has led you to believe that I agree with Hilbert.  Well, I
>>don't, as you can plainly see.  In truth, mathematics uses not a formal system,
>>but an enormously sophisticated symbolic system, namely language.

IS:
>The trouble with natural language is that it is inconsistent.

The trouble with the language of mathematics is that all valiant attempts
to divorce, or even consistently distinguish it from natural language have
ended in abject failure.

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

`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'
: Qu'est-ce qui est bien?  Qu'est-ce qui est laid?         Harvard   :
: Qu'est-ce qui est grand, fort, faible...                 doesn't   :
: Connais pas! Connais pas!                                 think    :
:                                                             so     :
: Mikhail Zeleny                                                     :
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