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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$1@dont-email.me>, "dilettante" <no@nonono.no> writes:
>"David C. Ullrich" <ullrich@math.okstate.edu> wrote in message news:c3qh38lilvho3lnar8gvo1po7rbhmokflr@4ax.com...
>> On Fri, 24 Aug 2012 20:14:13 +0100, Frederick Williams <freddywilliams@btinternet.com> 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
#include <Standard_Disclaimer>
Life's too important to take seriously.



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