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Re: unable to prove?
Posted:
Aug 27, 2012 1:23 PM
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dilettante 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 > >> >>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.
Nor me! Note that G\"odel's constructible universe is a model of CH (and all the axioms of set theory), while various Cohen model's are models of not-CH (and all the axioms of set theory). Are any of these models _the_ universe of sets? I have no idea. What is the universe of sets?
> I thought my remarks would provoke more discussion.
Let's hope they do.
> 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.
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