|
|
Re: unable to prove?
Posted:
Aug 27, 2012 3:29 PM
|
|
"David C. Ullrich" <ullrich@math.okstate.edu> wrote in message
news:2v6n38llcvc1t2jb6iidldij9mrg2ieflp@4ax.com...
> On Mon, 27 Aug 2012 09:00:20 -0500, "dilettante" <no@nonono.no> 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.
>>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.
>
> I read your post. I started thinking about a reply, came to the
> conclusion that explaining clearly why something is not clear
> to me would be difficult...
>
> Far from a complete answer: How can there be any
> confusion over the status of the power set of X?
> It's simply
>
> P(X) = {y : y subset X}.
>
> What could possibly go wrong with that? Well, the
> question is why there _is_ a _set_ S with the
> property that for every y, y is in S if and only
> if y is a subset of X.
>
> There's also no problem with
>
> R = {x : x is not an element of x},
>
> except of course there _are_ problems with that.
Right. Positing the existence of that set leads to a contradiction, so ZFC
and related systems bar the existence of such a set via the concept of
proper classes or something. I suppose it isn't known whether positing the
existence of the power set of the reals leads to a contradiction. Surely
also taking the existence of this set as an axiom doesn't do anything to
resolve CH (of course, as far as anyone knows - if it leads to a
contradiction it resolves everything, in a sense)? (Yes, that's a question)
>
> Not an answer, really, just an illustration of how
> it can be that things are much less clear than
> they seem at first.
Thanks nevertheless: it's nice to know that this isn't an easy thing to
explain, even for someone who knows a lot more mathematics than I do.
>
>>>
>>> --
>>> The animated figures stand
>>> Adorning every public street
>>> And seem to breathe in stone, or
>>> Move their marble feet.
>
|
|
