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?
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