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From: veenstra@uiucdcs.UUCP (veenstra )
Newsgroups: net.math
Subject: 'Choose' your axiom - (nf)
Message-ID: <3145@uiucdcs.UUCP>
Date: Thu, 6-Oct-83 23:27:41 EDT
Article-I.D.: uiucdcs.3145
Posted: Thu Oct  6 23:27:41 1983
Date-Received: Sat, 8-Oct-83 03:08:02 EDT
Lines: 37

#N:uiucdcs:28200023:000:1638
uiucdcs!veenstra    Oct  6 22:11:00 1983


One of the most interesting axioms of set theory is the Axiom of Choice:

Given any (possibly uncountable) collection of non-empty sets, there exists
another set containing one element from each set.

Intuitively, you can 'choose' an element from each set (hence the name "Choice
Axiom"). For example, from an infinite collection of pairs of socks, one can
choose one sock from each pair. It is well known that one can also 'choose'
to accept the "Choice Axiom" or reject it.  In either case, one would still
be consistent with the rest of the axioms of set theory.

Although the theorem seems intutitively true, one can derive strange and
counter-intuitive results from it.  I came across one of these strange results
in a book which a friend of mine pointed out to me.  It goes something like
the following:

Imagine two spheres, one very large, like the sun, and one very small, like
a pea.  The entire sphere is being referred to here, not just the surface.
One can partition each sphere into finitely many pieces, say n. Each sphere
is partitioned into n pieces, giving 2n pieces altogether. Assuming the Axiom
of Choice is true, it is possible to partition the spheres in such a way
that each piece of the large sphere is congruent to a (distinct) piece of the
small sphere.

I have never been able to find a proof of this result. (The fact that I would
probably not understand the proof does not prevent me from wanting to see it!)
Has anyone seen a proof of this?  Are there any other counter-intuitive results
that you know of that result from assuming the Axiom of Choice?

You can mail to ...uiucdcs!veenstra.


       Jack