From: hrubin@odds.stat.purdue.edu (Herman Rubin) Subject: Re: banach axiom Date: 29 Jun 1999 11:15:58 -0500 Newsgroups: sci.math Keywords: Axiom of Choice, Banach-Tarski result, others: implications? In article <37659C9F.3C45B08B@student.canterbury.ac.nz>, Ayan Mahalanobis wrote: >David C. Ullrich wrote: >> > Lets come to a point, is there a non-lebesgue measurable subset in >> R^3. >> Yes, at least in the most standard setup. >> > if so what you need to assume to prove that? >> The axiom of choice. (Things like the Banach-Tarski paradox >> sometimes motivate people to want to throw out AC. But it's known >> that AC does not cause any actual inconsistencies - if standard >> set theory without AC is consistent then it's still consistent >> when AC is added to the axioms.) >Technically you are true, that if ZF is consistence then so is ZFC, >ZFC+AC, ZFC+~AC(not to sure can't remember of any reference). But what >about meaning in mathematics. Banach Taraski paradox seems to be a >meaningless proposition to me. Its equivalence with AC forces me to >think AC to be a meaningless proposition. Of course these are my >choices. Banach-Tarski is NOT equivalent to the Axiom of Choice. I am not sure where it stands in the hierarchy, but it has been proved from the Hahn-Banach Theorem, which is weaker than the Boolean Prime Ideal Theorem, which is weaker than the Axiom of Choice. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399 hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558