Michael Stemper wrote: > > 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. > =================================================================
Just to muddy the waters(*): in second order ZF, CH is provable or refutable. Nobody knows which unless they can settle CH one way or the other first.
-- The animated figures stand Adorning every public street And seem to breathe in stone, or Move their marble feet.