Re: induction vs Cantor

From: Daryl McCullough (daryl_at_atc-nycorp.com)
Date: 12/03/04 

Date: 3 Dec 2004 04:03:31 -0800

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.

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

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