markp@tekmdp.UUCP (Mark Paulin) (10/07/83)
My favorite formulation of the "Axiom of Choice" is: "The cartesian product of a nonempty family of nonempty sets is nonempty." The counter-intuitive result mentioned is the so-called "Banach-Tarski Paradox" which is sometimes stated as, "a sphere the size of a pea may be cut into finitely many pieces which may be assembled into a life-size statue of Banach." It has been shown that the axiom of choice is required to prove the existence of a non-lebesgue-measureable set. To me, acceptance/rejection of the axiom hinges on the difference between what *can* be done and what *may* be done. There may be *no way* to actually find the cartesian product in the formulation above (or the "choice set" in the other formulation) but that does not mean (to me) that these sets do not exist, so I accept the axiom. Mark Paulin ...tektronix!tekmdp!markp
leichter@yale-com.UUCP (Jerry Leichter) (10/09/83)
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
dmn@uvacs.UUCP (10/09/83)
References: tekmdp.2286 Some time ago I attended a talk in which the proof of the Banach-Tarski theorem was presented. One step in the partitioning of the pea sphere involves picking points from the sphere that aren't *close* to each other in real space (using the Axiom of Choice) and calling this collection a *piece* of that sphere. A mathematical point has no volume so that to my mind the counter-intuitive result follows more from the way this so called piece is defined than from the fact that the Axiom of Choice is used in the proof. David Nicol (uvax/dmn)