
Re: unable to prove?
Posted:
Aug 28, 2012 8:20 AM


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...
>> >> 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. >>

