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

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Posts: 141
Registered: 5/15/12
Re: unable to prove?
Posted: Aug 27, 2012 6:15 PM
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"Michael Stemper" <> wrote in message
> In article <k1aqov$3b9$>, "dilettante" <>
> writes:

>>"David C. Ullrich" <> wrote in message

>>> On Fri, 24 Aug 2012 20:14:13 +0100, Frederick Williams
>>> <> wrote:

>>>>What about the continuum hypothesis in place of RH?
>>> In my opinion (with which many diisagree) it's not clear that CH
>>> _is_either true or false in any absolute sense. If so then it's
>>> much more problematic here.

>> 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
>>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.

> --
> Michael F. Stemper
> #include <Standard_Disclaimer>
> Life's too important to take seriously.

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