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Re: unable to prove?
Posted:
Sep 2, 2012 7:11 PM
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On 2012-09-02, David Bernier <david250@videotron.ca> wrote:
> On 08/28/2012 08:53 AM, Michael Stemper wrote:
>> In article<k1gri5$k9d$1@dont-email.me>, "dilettante"<no@nonono.no> writes:
>>> "Michael Stemper"<mstemper@walkabout.empros.com> wrote in message news:k1g8tg$jql$1@dont-email.me...
>>>> In article<k1aqov$3b9$1@dont-email.me>, "dilettante"<no@nonono.no> 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
hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
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