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Topic: unable to prove?
Replies: 28   Last Post: Sep 18, 2012 3:54 PM

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David Bernier

Posts: 3,892
Registered: 12/13/04
Re: unable to prove?
Posted: Sep 1, 2012 10:35 PM
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On 08/28/2012 08:53 AM, Michael Stemper wrote:
> In article<k1gri5$k9d$>, "dilettante"<> writes:
>> "Michael Stemper"<> wrote in message news:k1g8tg$jql$
>>> In article<k1aqov$3b9$>, "dilettante"<> 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

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