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