Re: induction vs Cantor

From: Virgil (ITSnetNOTcom#virgil_at_COMCAST.com)
Date: 12/03/04 

Date: Fri, 03 Dec 2004 11:58:03 -0700

In article <copkmj022nn@drn.newsguy.com>,
 daryl@atc-nycorp.com (Daryl McCullough) wrote:

> Poker Joker says...
> >
> >"Virgil" <ITSnetNOTcom#virgil@COMCAST.com> wrote
>
> >> what I have called the Cantor function from R^N to R.
> >
> >Describe this class of functions and show how Cantor used them in his
> >proof as you say. I'm looking for where he classified them as R^N to R
> >and that they were uncountable.
>
> I think Virgil means the class of functions f which given
> an infinite list of reals returns a new real that is not
> on the list.

Precisely!
>
> Cantor came up with only one such function: his diagonalization
> function. R^N just means the set of functions from N to R, or
> equivalently, the set of infinite lists of reals, or a set of
> reals indexed by the naturals. Cantor's diagonalization procedure
> defines a function which given an infinite list of reals (that is,
> an element of R^N) returns an element of R that is not on the list.
>
> Cantor didn't prove that there were uncountably many such functions,
> he only proved that there was one. That's all he needs to be able
> to show that no list of reals contains every real.
>
> It is easy enough to show that the cardinality of the set of such
> functions is uncountable, although this fact is not used in Cantor's
> proof.
>
> --
> Daryl McCullough
> Ithaca, NY

Thank you!

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