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

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Michael Stemper

Posts: 671
Registered: 6/26/08
Re: unable to prove?
Posted: Aug 27, 2012 12:57 PM
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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 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.

Michael F. Stemper
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