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Re: Objection to Cantor's First Proof
Posted:
Jul 25, 2012 10:10 AM
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On Wed, 25 Jul 2012 03:30:20 +0100, "LudovicoVan"
<julio@diegidio.name> wrote:
>[More a bunch of questions than any solid objection.]
>
><http://en.wikipedia.org/wiki/Cantor's_first_uncountability_proof>
You might reproduce this proof here, or at least state what's being
proved.
> << 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,... and any
interval [a,b], 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!
See, it's important to keep straight what's being proved.
The proof does _not_ begin by assuming that x_1,...
is an enumeration of the reals! If it _did_ begin with
that assumption then yes, the number of intervals
would be infinite.
>and not even how the limit could not be an improper
>interval (i.e. isn't it always a_oo = b_oo?).
Certainly not.
> << 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. >>
>
>Let [ a_n, b_n ] be the sequence of real intervals:
>
> [ a_0, b_0 ] := [ a, b ]
> [ a_n, b_n ] c [ a_{n-1}, b_{n-1} ], n > 0
>
>with:
>
> a_n := x_{n'[2n-1]}
> b_n := x_{n'[2n]}
>
>where x_{n'[2n-1]} and x_{n'[2n]} are, non respectively, the (2n-1)-th and
>(2n)-th entries picked from sequence x per the rules of the game.
>
>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]."
>
>From (1) the conclusion allegedly follows.
>
>OTOH, let's consider the limit interval:
>
> [ a_oo, b_oo ] =
> = lim_{n->oo} [ a_n, b_n ] =
> = lim_{n->oo} [ x_{n'[2n-1]}, x_{n'[2n]} ]
>
>Then, we also have the property:
>
> (2.1) a_oo = lim_{n->oo} a_n = lim_{n->oo} x_{n'[2n-1]}
> (2.2) b_oo = lim_{n->oo} b_n = lim_{n->oo} x_{n'[2n]}
>
>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
other than saying "it does not follow" and that it seems like
a trick that reminds you of something else.
_Since_ you're expressing confusing here instead of just
asserting that mathematicians are all wrong, I'll write out
the key points in somewhat more detail than in that
wikipedia article.
Don't tell me what things remind you of and how things
seem. DON'T jump to the end and talk about the
conclusion. Tell me the FIRST step below that you
don't follow:
Asssuming there are infinitely many intervals.
We let a_1 and b_1 be the two first x_j that lie
in the interior of [a,b]. Meaning that a < a_1 < b_1 < b.
To simplify things, let's assume that a_1 = x_25 and b_1 = x_23.
Now this follows:
(*) If x_j lies in the interior of [a_1,b_1] then j > 25.
Why? Because if x_j lies in the interior of [a_1,b_1]
and j <= 25 then the choice of a_1, b_1 would have been
different! We cant't have x_11 in [a_1,b_1], becuause
x_23 and x_25 were the first two x_j lying in [a,b].
Now we choose a_2 and b_2, etc. Maybe a_2 = x_300
and b_2 = x_4999. Now for the same reason as before,
it follows that
(*) 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]. So _if_ a_00 = x_j then j > 25.
And a_00 lies in the interior of [a_2,b_2].
So if a_00 = x_j then j > 4999.
Etc. a_oo cannot be x_j for any j, because 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.
>
>Thanks,
>
>-LV
>
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