
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 >>>> >>giraffelike 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 >>>> >>bugfree. >>>> >>(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. >

