From: hrubin@odds.stat.purdue.edu (Herman Rubin) Subject: Re: Axiom of Choice Date: 5 Jul 2000 10:50:18 -0500 Newsgroups: sci.math Summary: [missing] In article <8jus6e$qul$1@nnrp1.deja.com>, Jim Heckman wrote: >In article <8jujca$d07$2@cantuc.canterbury.ac.nz>, >mathwft@math.canterbury.ac.nz (Bill Taylor) wrote: .................. >> I hadn't been aware of this before, I'd thought that as soon as it >> was identified (Zermelo 1904?) it became immediately accepted. Can >> anyone enlighten us further on this matter? >At least one of my textbooks claims that, in fact, when Zermelo proved >the Well-Ordering Principle, mathematicians were so astonished that >they looked for holes in his proof, and found: AC. This was not the first explicit statement of AC. Russell is certainly earlier. >BTW, in ZF+DC, what can be said about the cardinality of the real >line R? It must still be Pow(aleph_0) =? 2^{aleph_0}, mustn't it? This does not require anything more than ZF. Some of us, including myself, have even posted an explicit 1-1 correspondence, using continued fractions. I'm >sure I've read that it's possible to prove that R is uncountable using >only its order properties, A complete dense linearly ordered set with no first or last element and a countable dense subset is isomorphic to the reals. Other conditions can be used. as opposed to the way it's usually done >using its algebraic properties, via infinite binary series, etc. The proof is not that much different. And >I think I've read it's possible to prove that Pow(c) (=? 2^c) > c for >any cardinality c, even without AC. This is essentially the Russell paradox. Does the existence of uncountable >well-ordered sets, which I *know* can be shown without full AC, depend >on DC? It depends on nothing beyond ZF. The Hartogs function, the set of all equivalence classes of well-ordered sets no larger than a given set, is not as small as that set. -- 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