In article <email@example.com>, "dilettante" <firstname.lastname@example.org> writes: >"David C. Ullrich" <email@example.com> wrote in message news:firstname.lastname@example.org... >> On Fri, 24 Aug 2012 20:14:13 +0100, Frederick Williams <email@example.com> wrote:
>>>What about the continuum hypothesis in place of RH? >> >> In my opinion (with which many diisagree) it's not clear that CH >> _is_either true or false in any absolute sense. If so then it's >> much more problematic here. > > This has always been a little disconcerting for me. I've read that it was >proved that CH is independent of the usual axioms of set theory, or >something like that. It seems to me that if the real numbers are a well >defined object, then its power set should be a well defined object, and it >should be the case that either some member of that power set has cardinality >between that of the naturals and that of the reals, or not. If such an >animal did exist, it should be at least possible for someone to exhibit it >in some way - "here it is, now what about that independence?"
I asked a very similar question here eighteen years back (give or take a month). One Mike Oliver responded:
================================================================= >Although CH is independent of ZF, isn't it still possible that >somebody could find a set that violates it?
It depends on what you mean by "find." It is not possible to define a set of reals and prove *in ZFC* that it has cardinality strictly between that of the integers and that of the real numbers.
But you might be able to define a set that "really" has this property, even though not provably in ZFC. =================================================================
-- Michael F. Stemper #include <Standard_Disclaimer> Life's too important to take seriously.