Re: Bedeviled by the .999~ = 1 "debate"

In article <denvmc$ssb$1@xxxxxxxxxxxxxxxxxxx>,
klewis@xxxxxxxxxxxxxxxx (Keith A. Lewis) wrote:

> Ittay Weiss <weiss@xxxxxxxxxx> writes in article
> <9951112.1125087940838.JavaMail.jakarta@xxxxxxxxxxxxxxxxxxxxxx> dated Fri, 26
> Aug 2005 16:25:10 EDT:
> >your proof is very nice (and, of course correct). However I fail to
> >understand
> >your remark about the AC. A great deal of mathematics depends on the AC. Of
> >course it is legitimate (and this is actually happening) no to accept the
> >axiom
> >of choice. One then gets different theorems...
> >
> >And what is wrong with the Banach-Tarski paradox???
>
> Taking apart a ball and re-assembling it into two complete balls, each the
> same size as the original, violates conservation of volume. I suspect
> that's why it's known as a paradox rather than simply a theorem. When you
> discover a paradox, you need to take a step back and re-think your axioms.

But no one claims that the pieces into which the original sphere is
disassembled are in any sense measurable sets of points.
.

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