From: Mike Oliver Subject: Re: The Sufficiency of the Axiom of Choice Date: Fri, 03 Aug 2001 02:35:09 -0700 Newsgroups: sci.math,sci.logic Summary: (Non-standard) cardinal E_0 Fred Galvin wrote: > > On Fri, 3 Aug 2001, Mike Oliver wrote: >> Well, it appears from Herman's example (which I now believe to be >> correct), that the existence of a nonmeasurable set follows >> from AC for E_0-many doubletons. A doubleton is certainly >> a countable set, and "E_0-many" implies that there's a surjection >> from R onto the index set. I'm not sure how all that comes >> out off the top of my head. > > What is the definition of E_0? E_0 is the equivalence relation on Baire space (functions from naturals to naturals) that says f E_0 g <==> (exists n)(all k>n) f(k)=g(k) (i.e. f and g are the same from some point on). By "E_0-many" I mean the cardinality of the quotient space of Baire space by this equivalence relation. Given AC, of course this is the same as the cardinality of the continuum, but in models of ZF+AD it's bigger. > Anyway, as far as I know (which isn't > saying much), constructions of nonmeasurable sets use more than > continuum many choices, which is why I asked. Does the axiom of choice > for all families of continuum many nonempty sets imply the existence > of nonmeasurable sets? I don't know, off the top of my head. I'll give it some thought.