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Re: unable to prove?
Posted:
Aug 28, 2012 8:20 AM
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On Mon, 27 Aug 2012 14:29:33 -0500, "dilettante" <no@nonono.no> wrote:
>
>"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)
My point is not that the power set axiom does or might lead to a
contradiction. The point is just that we really do need a power-set
axiom to say that the class of all subsets of X forms a _set_.
And once we're talking about sets that are given to us by
axioms it's no longer so clear that there is a "real" set out
there that the axioms are talking about.
I didn't put that very well. Like I said, I don't expect to put any
of this very well...
>>
>> 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.
>>
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