"Michael Stemper" <email@example.com> wrote in message news:firstname.lastname@example.org... > 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. > =================================================================
Interesting. I suppose the question of what this "really" consists of is one of those foundational questions that don't have an answer that is universally accepted.
> > > -- > Michael F. Stemper > #include <Standard_Disclaimer> > Life's too important to take seriously.