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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 10:00 AM
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"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.
I thought my remarks would provoke more discussion. 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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