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Re: unable to prove?
Posted:
Aug 27, 2012 1:23 PM
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dilettante wrote:
>
> "Frederick Williams" <freddywilliams@btinternet.com> wrote in message
> news:5038F0B2.8FA4186B@btinternet.com...
> > 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?
>
> Are you channelling WM? Somehow that doesn't clear the matter up for me.
Nor me! Note that G\"odel's constructible universe is a model of CH
(and all the axioms of set theory), while various Cohen model's are
models of not-CH (and all the axioms of set theory). Are any of these
models _the_ universe of sets? I have no idea. What is the universe of
sets?
> I thought my remarks would provoke more discussion.
Let's hope they do.
> Perhaps no one has
> anything to say about this, or perhaps not many read my posts. I suppose I
> could remedy the latter by getting either crazier or nastier, or go the
> arduous route of posting clever, interesting, and helpful stuff, but no -
> I'll just stay my mostly sane, not too horribly nasty, boring self, and be
> satisfied with the status quo.
--
The animated figures stand
Adorning every public street
And seem to breathe in stone, or
Move their marble feet.
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