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Re: unable to prove?
Posted:
Aug 28, 2012 9:30 AM
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"David C. Ullrich" <ullrich@math.okstate.edu> wrote in message
news:fmdp385tg6s42j77jq6g76adcojfui5btq@4ax.com...
> 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...
Well enough, but I would like to clarify one point: in the proof that CH is
independent, is the power set axiom is in effect?
In other words, taking the existence of the power set of the reals as a
given, is CH still proved to be independent?
,>>>
>>> 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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