"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? > >