On 2012-09-02, Herman Rubin <firstname.lastname@example.org> wrote: > 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: > >>>>>e 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
Here is an explicit encoding into P(P(rationals)), a modification of Hartog's original construction of the smallest aleph not less than or equal to the cardinality of a set.
Each countable ordinal can be represented as a set of rational numbers with their usual ordering. Thus it can be matched with the set of all such which represent it. This gives the explicit ordering.
Since P(rationals) is equivalent to R, this gives the desired embedding.
-- 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