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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 28, 2012 8:53 AM
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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
>>>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.

Michael F. Stemper
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