Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP
Path: utzoo!linus!decvax!yale-com!leichter
From: leichter@yale-com.UUCP (Jerry Leichter)
Newsgroups: net.math
Subject: Re: Axiom of Choice
Message-ID: <2140@yale-com.UUCP>
Date: Sat, 8-Oct-83 18:45:12 EDT
Article-I.D.: yale-com.2140
Posted: Sat Oct  8 18:45:12 1983
Date-Received: Sun, 9-Oct-83 12:21:25 EDT
References: tekmdp.2286
Lines: 32

I discussed the Banach-Tarski paradox in this newsgroup a couple of months
back.  As I recall, someone also submitted a fairly elementary proof, or
at least an outline of one.  It's actually not at all hard; I saw such a
proof once, years ago, but don't recall it.

Just to make things more definite:  It is possible to cut a (solid) sphere of
arbitrary size into 5 pieces, and reassemble the pieces to from 2 spheres
each of the same size (radius) as the original.  (It turns out that 4 pieces
are sufficient if you start with a sphere missing its center point; the two
new ones will also be missing their center points.)  The "way it works", for
the 4-piece case, is that you get 4 pieces A, B, C and D with the odd property
that A and B are each congruent to A+B - "congruent" in the good old Euclidean
sense; same for C, D and C + D.  I don't remember what the 5th piece does in
the "whole sphere" case.

The proof for 5 pieces is involved.  The simple proof uses 11 pieces.

Before you think this is a way to solve the energy crisis - by multiplying
pieces of coal or whatever - be aware that the pieces are made by "cuts"
something vaguely like:  Put all pieces one of whose coordinates is a
rational number and one of whose ^points^ coordinates is an irrational in
A; etc. - i.e. you have to divide the spheres up literally "point by point".

BTW, Banach-Tarski does NOT work in two dimensions; however, in two and even
in one dimension we have another result:

The existence of a countably additive, translation-invarient measure (an "area"
in two dimensions, a measure of length in one; also, in one, a probability
measure, if defined right) on ALL subsets of the line (or plane) is equivalent
to the axiom of choice.
							-- Jerry
					decvax!yale-comix!leichter leichter@yale