veenstra@uiucdcs.UUCP (veenstra ) (10/07/83)
#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.
Jackjames@umcp-cs.UUCP (10/08/83)
1) What is the book you found this in? 2) Why do you say this is counter-intuitive? It seems to me that if I divide up two spheres in the same "way", then the pieces will necessarily be in one-to-one congruence/correspondence. The only difference between corresponding pieces will be their sizes, which should be in the same ratio as the original spheres. --Jim
veenstra@uiucdcs.UUCP (10/09/83)
#R:uiucdcs:28200023:uiucdcs:28200024:000:358
uiucdcs!veenstra Oct 8 23:54:00 1983
1) I can't recall the name of the book but I will try to find out.
2) You are mistaking 'congruence' for 'similarity'. Just as congurent
triangles have equal angles and sides, so the 3-dimensional pieces
have the exact same volume. I think you will agree that the result
is now counter-intuitive.
Jack (at uiucdcs)