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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 25, 2012 11:35 AM
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dilettante wrote:
>
> "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:
> >

> >>TS742 wrote:
> >>>
> >>> Are some hypotheses unprovable?

> >>
> >>Idiots like me may say, "no, the hypothesis 0 =/= 0 is unprovable".
> >>
> >>Do you mean "are some truths unprovable?"? I don't know. Some may
> >>claim that the truths of mathematics ae just those statements that are
> >>provable.
> >>

> >>> Or do they all have a proof that is
> >>> just not found yet? The Riemann hypothesis comes to mind.

> >>
> >>Let's suppose that RH is true. "RH is unprovable" may mean various
> >>things:
> >>(1) Humans could prove it were it not for the fact that they will become
> >>extinct before they do so. (And that "could" means what?)
> >>(2) Humans can't prove it because their brains are too feeble. (But the
> >>giraffe-like beings on planet Scorrrf (my keyboard doesn't have the
> >>diacritics that the first and third "r"s should have) prove it as
> >>homework in their first year a school.)
> >>(3) A computer (built and programmed by another computer) proved it
> >>after running for sixty years, but no one is foolhardy enough to claim
> >>that they understand what that computer is doing or that it is bug-free.
> >>(4) No machine or creature in this universe or any other will ever prove
> >>it.
> >>
> >>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?" The fact that
> this isn't so is very strange to me, but there are more things in heaven and
> earth than are dreamt of in our philosophy, Horatio.
> Any thoughts on how to better grasp this little conundrum?


What is the set of _all_ subsets of a set X? If X is finite, the
question is easily answered by listing them, but otherwise?

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



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