On 2012-09-02, David Bernier <email@example.com> wrote: > 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 ...
> David Bernier
I do not believe it is possible to get one. I believe the non-existence of such an encoding is consistent with the axioms of set theory.
-- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University firstname.lastname@example.org Phone: (765)494-6054 FAX: (765)494-0558