In article <email@example.com>, "dilettante" <firstname.lastname@example.org> writes: >"Michael Stemper" <email@example.com> wrote in message news:firstname.lastname@example.org... >> In article <email@example.com>, "dilettante" <firstname.lastname@example.org> writes:
>>> 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.
That would be my guess, as well.
-- Michael F. Stemper #include <Standard_Disclaimer> Life's too important to take seriously.