On 08/28/2012 08:53 AM, Michael Stemper wrote: > In article<firstname.lastname@example.org>, "dilettante"<email@example.com> writes: >> "Michael Stemper"<firstname.lastname@example.org> wrote in message news:email@example.com... >>> In article<firstname.lastname@example.org>, "dilettante"<email@example.com> 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. >
From what I remember, omega_1 is used in set theory to denote the set of countable ordinals.
I tried to connceive of a system, an encoding, which associated to every countable ordinal alpha a subset of the real numbers in a unique way.
The idea was to get an injection j: omega_1 -> P(R), R =real numbers.
(one could try with P(Q) or P(N) also).
Preferably, each alpha would be mapped to a countable set of reals.
I never found any formulaic, explicit encoding ...