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

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dilettante

Posts: 141
Registered: 5/15/12
Re: unable to prove?
Posted: Aug 27, 2012 6:15 PM
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"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:

>>"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.
> =================================================================


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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