"Frederick Williams" <firstname.lastname@example.org> wrote in message news:5038F0B2.8FA4186B@btinternet.com... > dilettante wrote: >> >> "David C. Ullrich" <email@example.com> wrote in message >> news:firstname.lastname@example.org... >> > On Fri, 24 Aug 2012 20:14:13 +0100, Frederick Williams >> > <email@example.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. > > -- > The animated figures stand > Adorning every public street > And seem to breathe in stone, or > Move their marble feet.