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

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Frederick Williams

Posts: 2,164
Registered: 10/4/10
Re: unable to prove?
Posted: Aug 28, 2012 11:04 AM
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Michael Stemper wrote:
> 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.
> =================================================================

Just to muddy the waters(*): in second order ZF, CH is provable or
refutable. Nobody knows which unless they can settle CH one way or the
other first.

(* 1913?--1983)

The animated figures stand
Adorning every public street
And seem to breathe in stone, or
Move their marble feet.

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