
Re: unable to prove?
Posted:
Aug 28, 2012 9:30 AM


"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 >>>>>> >>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) > > 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 powerset > 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. >>> >

