From: rld@math.ohio-state.edu (Randall Dougherty)
Subject: Re: Banach-Tarski
Date: 22 Oct 1999 15:54:05 GMT
Newsgroups: sci.math
Keywords: Comparison of Banach Tarski theorem and Axiom of Choice
In article ,
Fred Galvin wrote:
>Actually, you *can* assign an "area" of sorts to every subset of the
>*plane*. This is called a "Banach measure": the measure is invariant under
>Euclidean motions, it is finitely (but not countably) additive, and every
>set is measurable. The existence of Banach measures proves the
>impossibility of a Banach-Tarski Paradox in 2 dimensions.
>
>Bu the way, I *think* you need the Axiom of Choice in order to prove the
>existence of Banach measure. Maybe one of the set-theory experts who reads
>this bulletin board can tell me if a planar Banach-Tarski Paradox is
>consistent with ZF? Ilias?
According to chapter 13 of Wagon's "The Banach-Tarski Paradox,"
the nonexistence of a Banach measure is consistent with ZF+DC
(in fact, Pincus and Solovay constructed a model of ZF+DC in which
a much more general measure nonexistence statement holds),
but the nonexistence of a planar Banach-Tarski paradox is
provable in ZF alone (because one only needs a countable part
of a Banach measure to show that a given decomposition can't work).
Randall Dougherty rld@math.ohio-state.edu
Department of Mathematics, Ohio State University, Columbus, OH 43210 USA
"I have yet to see any problem, however complicated, that when looked at in the
right way didn't become still more complicated." Poul Anderson, "Call Me Joe"