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!europa.eng.gtefsd.com!howland.reston.ans.net!pipex!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>
Reply-To: rbj@campion.demon.co.uk
X-Newsreader: Demon Internet Simple News v1.29
Lines: 73
Date: Tue, 25 Oct 1994 05:51:22 +0000
Message-ID: <783064282snz@campion.demon.co.uk>
Sender: usenet@demon.co.uk

In article <38653l$ivf@mp.cs.niu.edu> rickert@cs.niu.edu "Neil Rickert" writes:

> In <385i1s$69h@toves.cs.city.ac.uk> jampel@cs.city.ac.uk (Michael Jampel)
>  writes:
> 
> >More background: the Axiom of Choice (AC) is well-known because, unlike
> >most other axioms, it is not definitely clear that its truth is
> >`self-evident'. It seems that it _should_ be true, because otherwise we
> >cannot prove e.g. that every infinite vector space has a basis, which
> >is something that would appear to be true/likely.
> 
> The idea that axioms should be self evident has long passed out of
> the traditions of mathematics.  The problems with the parallel
> postulate were already enough to lead to reject of the 'self-evident'
> criterion.

This is not quite correct.  It is true that mathematicians are now content
to work with fairly arbitrary axiomatic systems, provided that they can
assure themselves that they are consistent, i.e. have at least one model.
However, this is because they work within the more general framework of
classical set theory in which judgements about consistency of systems
like geometry can be made.

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.  The result is much greater uncertainty in the case of NF about
(1) whether the system is consistent, and (2) whether the system is an
appropriate foundation for mathematics.

> >My question: is the Axiom of Choice TRUE.
> 
> Of course the Axiom of Choice is TRUE.  In mathematical systems,
> axioms are true BY DEFINITION.  It is true in the sense of truth
> within a formal mathematical system.  It is not intended to say
> anything about the real world.

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.

Addendum
========

I joined all this Penrose stuff very late, so I don't know whether this has
been said.

I believe that his argument is based on the premise that:

	mathematical results obtained by a computer program must be
	the theorems of a consistent semi-decidable formal system

This is probably a mistake arising from misunderstanding the consequences
of the fact that computers compute according to algorithms.  In fact
the two are close, a set generated by a computer without external influences
must be semi-decidable.  But it need not be "consistent".  And even
artificial intelligentsia may be allowed discourse with the external world.

If he doesn't take this view then I can't see why he should consider
Goedel's result applicable.  

Since this premise is false, so may be the conclusion.

Other authors have also drawn conclusions about AI while implicitly
accepting this assumption (I believe Hofstater does).

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