"David C. Ullrich" <
ull...@math.okstate.edu> wrote in message
news:cbuv08t9odnda4gva...@4ax.com...
<snipped>
> > <
http://en.wikipedia.org/wiki/Cantor's_first_uncountability_proof>
> > << Cantor now breaks the proof into two cases: Either the
> > number of intervals generated is finite or infinite. >>
> > Given the assumption of completeness, I cannot see how the number
> > of intervals can be finite,
>
> What's being proved is this: Given any sequence x_1,...
Given any sequence (x_n) *of real numbers*. (I would not see what else,
anyway so it is stated by Wikipedia, too.)
> and any interval [a,b],
Again, any interval [ a, b ] of real numbers.
> there exists an element of [a,b] which is not one of the x_j.
>
> The number of intervals certainly can be finite. At each stage of
> the proof we choose the first two x_j's which are elements of
> [a_n,b_n] and use them for the endpoints of the next interval.
> But there's no reason that [a_n,b_n] has to contain any x_j
> at all!
I thought that must be the case because we are dealing with real numbers: in
fact the completeness property is assumed in the proof, for the limits to
exist.
> The proof does _not_ begin by assuming that x_1,...
> is an enumeration of the reals!
Unless I am missing something, it definitely does.
> If it _did_ begin with
> that assumption then yes, the number of intervals
> would be infinite.
OK.
> > and not even how the limit could not be an improper
> > interval (i.e. isn't it always a_oo = b_oo?).
>
> Certainly not.
Hence, unless I am missing something, certainly so.
> > << If the number of intervals is infinite, let a_oo = lim_{n->oo}
> > a_n.
> > At this point, Cantor could finish his proof by noting that a_oo is not
> > contained in the given sequence since for every n, a_oo belongs to the
> > interior of [a_n, b_n] but x_n does not. >>
> > Then, we have the property:
> >
> > (1) E m : A n : n>m -> ~ ( x_n e [ a_n, b_n ] )
> >
> > i.e. the property that "for every n, x_n does not belong to the interior
> > of
> > [a_n, b_n]."
Later corrected to:
(1) E m : A n : n>m -> ~ ( x_n e ] a_n, b_n [ )
> > The main objection is that the thesis does *not* follow from (1): I
> > think
> > the conclusion amounts to the same kind of invalid reasoning found in
> > the
> > standard solution to the balls and vase problem, i.e. "incorrect
> > counting".
> > Moreover, from property (2) and (I suppose) completeness, I think we are
> > in
> > fact showing that a_oo (or, b_oo) get picked up from sequence x. --
> > Roughly speaking, the proof seems to amount to a "trick with indexes".
> > Otherwise, could anyone formalize the last step of the proof, i.e. the
> > conclusion, to see which derivation is actually at play?
>
> You don't give any explanation for what you think is wrong
Actually, you do not formalize the conclusion, while I do have posted a more
precise objection, which was this:
"A n, ~ ( x_n e [ a_n, b_n ])"
does *not* contradict:
"A n, a_n (or, b_n) is an element of sequence x".
That I think exposes the alleged "trick" (a paralogism quite similar to the
"every
ball is eventually removed, hence the vase must end up empty" of the
Ross-Littlewood paradox).
But that objection needs be corrected (as mentioned in another post, too),
because the actual property from which the conclusion allegedly follows is:
(1) E m : A n : ( n > m ) -> ~ ( x_n e ] a_n, b_n [ )
So, I will have to try and work out a correct formalization of the objection
(which, as above, is going to be of the same kind of the "for every ball
removed many more get in, so that the vase is never empty").
> Asssuming there are infinitely many intervals.
> [...]
>
> (*) If x_j lies in the interior of [a_2,b_2] then j > 4999.
>
> Etc.
>
> Now take that limit. The number a_oo _does_ lie in
> [a_1.b_1]. [...]
Yes: as I get it, by completeness, the limit exists and, by construction, it
is internal to all intervals at finite steps (i.e. A n : a_oo e ] a_n, b_n
[), this being a sequence of strictly nested intervals.
> So _if_ a_00 = x_j then j > 25.
> Etc. a_oo cannot be x_j for any j, because
No, I think a_oo cannot be equal to any x_n for n in N because, by
definition, a_oo := lim_{n->oo} a_n, then by the properties of completeness
and of the construction just mentioned above.
> if
> a_oo = x_j then j > 25 by (*) and also j > 4999
> by (**), and also j > 10000 by (***), the first
> step I left out. The number j would have to be
> larger than infinitely many natral numbers, and
> there is no such natural number j.
I'd contend your "induction" is as invalid as the paralogism indicated
above. Namely, you are saying that we "run out of indexes", or something to
that effect. Anyway, that is not a formalization of the conclusion, it is
only a reworded informal presentation of it: I'd need it formally.
-LV