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Topic: unable to prove?
Replies: 28   Last Post: Sep 18, 2012 3:54 PM

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David C. Ullrich

Posts: 21,553
Registered: 12/6/04
Re: unable to prove?
Posted: Aug 27, 2012 12:14 PM
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On Mon, 27 Aug 2012 09:00:20 -0500, "dilettante" <> wrote:

>"Frederick Williams" <> wrote in message

>> dilettante wrote:
>>> "David C. Ullrich" <> wrote in message

>>> > On Fri, 24 Aug 2012 20:14:13 +0100, Frederick Williams
>>> > <> 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.

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.

Not an answer, really, just an illustration of how
it can be that things are much less clear than
they seem at first.

>> --
>> The animated figures stand
>> Adorning every public street
>> And seem to breathe in stone, or
>> Move their marble feet.

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