Newsgroups: comp.ai.philosophy
From: rbj@campion.demon.co.uk (Roger Bishop Jones)
Path: cantaloupe.srv.cs.cmu.edu!das-news2.harvard.edu!news2.near.net!news.mathworks.com!hookup!news.moneng.mei.com!howland.reston.ans.net!news.sprintlink.net!demon!campion.demon.co.uk!rbj
Subject: Re: Penrose & Banach-Tarski/Axiom of Choice
References: <385i1s$69h@toves.cs.city.ac.uk> <38653l$ivf@mp.cs.niu.edu> <783064282snz@campion.demon.co.uk> <38j2o5$8fm@mp.cs.niu.edu>
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Date: Sat, 29 Oct 1994 19:49:19 +0000
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In article <38j2o5$8fm@mp.cs.niu.edu> rickert@cs.niu.edu "Neil Rickert" writes:

> In <783064282snz@campion.demon.co.uk> rbj@campion.demon.co.uk (Roger Bishop
>  Jones) writes:
> >When we come to the axiomatisation of set theory things are slightly
> >different.  The reason why ZFC or NBG are accepted formalisations of
> >classical set theory rather than Quine's NF is that the former are plausibly
> >true of "the iterative conception of set", while the latter clearly is not,
> >and lacks any other semantic intuition against which we can judge the truth
> >of the axioms.
> 
> Generally speaking, mathematicians want their axioms to at least be
> plausible.  But I think it would be a mistake to assume that
> mathematicians accept ZFC over NF.  Most mathematicians do not
> concern themselves greatly with the details of the axiomatization of
> set theory, and have never considered the question.

Much mathematics can be done with indifference to particular axiomatisations
of set theory provided it is understood that the domain of discourse is
the classical iterative heirarchy.  The reason why indifference is possible
is precisely because axiom systems like ZFC, NBG,... have as common models
late enough stages in the construction of the iterative heirarchy and
therefore can consistently be combined.

The same cannot be said for NF.  Any mathematician who is confused about
whether he is working in the iterative heirarchy or in a model of NF is too
seriously confused to be trusted to produce sound mathematics, since these
two domains of discourse are radically incompatible (i.e. they have no models
in common).  A proof of Fermat's last theorem produced in a hybrid of ZFC
and NF could be made very short indeed.  If mathematicians don't concern
themselves with the difference between ZFC and NF it is because they don't give
any consideration to NF at all.

...

> >The question should perhaps be understood: "is the axiom of choice true
> >of the interpretation of set theory in the iterative conception of sets".
> >I happen to think that this is pretty obvious as well, but others 
> >(George Boolos, for example) are on record as doubting that it is determined
> >by that semantic.
> 
> I guess I would side with Boolos on that.  Our intuition about set
> theory is based on finite sets and constructible sets.  AC is quite
> plausible provided we restrict it to such sets.

Actually, Boolos's own account of the intutions behind set theory in the
iterative conception of set go well beyond finite and constructible sets.
It's a mystery to me, in the context of his description of these intuitions,
how he has difficulty in accepting that they entail the axiom of choice.

-- 
Roger Jones
rbj@campion.demon.co.uk
