Virgil
Posts:
3,521
Registered:
12/6/04
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Re: Grrrr! (was): infinite dimensional vector space
Posted:
Jun 26, 2003 1:26 AM
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In article <bddro9$59u$1@news.Stanford.EDU>,
Michael Hochster <michael@rgmiller.Stanford.EDU> wrote:
> In sci.math Kent Paul Dolan <xanthian@well.com> wrote:
>
> :> And even if the above were correct, here things get a little phony:
> :> That well-known diagonalization proof is exactly the proof that the
> :> reals are _uncountable_,
>
> : Nope. It is a proof that at least _one_ irrational number exists;
> : that the reals are uncountable doesn't have to be part of the initial
> : claim.
>
> Isn't "0.01001000100001... " a proof that at least one irrational number
> exists?
There are easier proofs of irrationals existing. What
"0.01001000100001... " proves is the existence of
at least one transcendental.
The Cantor diagonal proof proves that there can be n
surjection from the naturals to the reals.
This was not Cantor's first proof that there can be no such
surjection, but the diagonal proof is considerably less
technical, and much easier for a non-mathematician to
comprehend. The other proof relies on the completeness
properties of the reals rather than any representation in
decimal or other format.
Roughly, given that a bijection from the naturals to the
reals were to exist one can construct an increasing sequence
(I.S.) of reals and a decreasing sequence (D.S.) of reals so
that the I.S. is bounded above by each value in the D.S. and
the D.S. is similarly bounded below by each member of the
I.S., from which one may deduce that the greatest lower
bound proerty and least upper bound property of the reals
can not hold, so the bijection cannot exist after all.
This, with some of Cantor's other work on transfinites,
suffices to prove that the cardinality of the reals is
greater than the cardinality of the naturals.
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