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Re: Axiom of choice and the three spheres.
Posted:
Dec 1, 2006 7:17 PM
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In article
<ik60n2pcg859audb1p87ki0e5lfot54caq@4ax.com>,
David C. Ullrich <ullrich@math.okstate.edu> wrote:
> On Thu, 30 Nov 2006 22:59:00 GMT, Michael Press <jack@abc.net> wrote:
>
> >In article
> ><nvktm214s2m3vkep3lufj6s88etpnho36u@4ax.com>,
> > David C. Ullrich <ullrich@math.okstate.edu> wrote:
> >
> >> On Thu, 30 Nov 2006 01:49:23 GMT, Michael Press <jack@abc.net> wrote:
> >>
> >> >The decomposition of a sphere into pieces and assembly
> >> >of the pieces into two spheres congruent to the first
> >> >in [1] is often called the Banach-Tarski paradox. I
> >> >used that phrase to find the reference to their paper.
> >> >Yet their construction is not a paradox. It is not a
> >> >contradiction either logically or intuitively.
> >>
> >> It's lucky that the question of whether something
> >> is "intuitive" is not well-defined...
> >>
> >> >Reason:
> >> >we have no right to expect that disassembling a
> >> >measurable set A into non-measurable sets, then
> >> >reassembling them into a measurable set B implies that
> >> >mu(B) = mu(A). Not having read the paper, for all I
> >> >know their purpose was to give an example of this.
> >> >
> >> >To mount my hobby horse: the Axiom of Choice should not
> >> >be the whipping boy of every disagreeable result in set
> >> >theory. There is usually a better explanation for
> >> >problems than blaming the axiom of choice, if more
> >> >difficult to identify. Thanks.
> >>
> >> I'm missing your point here. First, I don't see people
> >> "blaming" AC for Banach-Tarski, nor saying that BT
> >> is "disagreeable"...
> >
> >Folks often delineate when they invoke AC. I keep
> >getting the impression that AC wears the scarlet
> >letter.
>
> Perhaps, but in my impression only among people
> who really don't understand the issues.
>
> For example, I know of various texts in analysis
> that purport to always say so when they use AC,
> but they nonetheless use the fact that a countable
> union of countable sets is countable without
> any special comment - seems very clear that these
> authors are unaware of the fact that _that_ result
> requires AC, and that if they did realize this
> they'd also realize that AC was their friend...
>
> >When analysts started constructing continuous
> >nowhere differentiable functions others found the
> >results disagreeable. I believe `monsters' came into
> >the discussion.
>
> Huh? What does _that_ have to do with AC? You
> simply define f to be a certain sum...
>
> One might also comment that people _found_ this
> disagreeable - if we're talking about how people
> feel about things at present that seems irrelevant.
>
> >> Anyway, without AC you can't prove the existence of
> >> a non-measurable set, hence you certainly can't prove
> >> BT. So if we _were_ going to assign "blame" for BT
> >> I don't see how AC gets off the hook so easily.
> >
> >Yes, I went off track there. Without AC all subsets of
> >R^n are measurable?
>
> I didn't say that. It's _consistent_ with ZF that
> all sets are measurable.
>
> >That is a convenient realm for
> >proving things; no worries about that hypothesis. :)
>
> I don't see it as being all that convenient, for example
> we lose the fact that a countable union of countable
> sets is countable.
>
> What seems to _me_ to be the practical significance
> of the fact that AC is needed to show that a
> non-measurable set exists is this:
>
> Roughly speaking, when you're trying to prove something
> in analysis showing that whatever is measurable is the
> last thing you should worry about - actually proving
> it may be a head-scratcher, but it's at least an empirical
> fact that it never seems to be what goes wrong. The fact
> that AC is needed to show that there exists a non-measurable
> set lends support to the idea that proving measurability is
> the last thing you should worry about.
>
> >> >[1] Banach and Tarski, Sur la décomposition des
> >> >ensembles de points en parties respectivement
> >> >congruentes", Fundamenta Mathematicae, 6, (1924),
> >> >244-277.
Thank you.
--
Michael Press
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