Re: Zenkin's paper on Cantor

From: Ross A. Finlayson (raf_at_tiki-lounge.com)
Date: 09/27/04 

Date: 27 Sep 2004 10:56:02 -0700

I think that it is wothwhile to consider Cantor's arguments with
regards to the nature of the real numbers, and their use.

Seen from different perspectives, the real numbers actually vary in
their composition. It's like the blind men and the elephant, except
different: the elephant is the snake, hippo, and zebra. In this case
the elephant is an elephant, but if you need a snake it'll do.

My arguments that I can biject the naturals and reals are somewhat
different than Zenkin's, I don't think I've ever argued that the
expansion was finite, but many or most other lines of argument, some
bad, some good, some dependent on other conditions, have been
addressed over the years here on sci.math math among us. I don't
validate Zenkin's argument at issue here.

When I was first asked to provide a bijection between the naturals and
reals, I replied with the Equivalency Function, so named because its
purpose is to exhibit a bijective mapping between the naturals and
unit interval of reals, among other things.

That's working well not only in the face of the antidiagonal argument,
but also Cantor's first proof, about which I say between any two
definite rational endpoints at some finite iteration there exists a
pair of irrationals, between which is, you guessed it: a rational.
Anyways, EF introduces the notion of iota, which is basically defined
as the smallest positive real number, that is basically Leibniz' dx
from the integral calculus. Neither the antidiagonal argument nor the
nested intervals arguments apply to EF, except perhaps in showing that
EF is the only way to biject N and R[0,1].

http://groups.google.com/groups?q=antidiagonal+sorted

As well, the reals are discussed vis-a-vis Dedekind cuts, Cauchy
sequences, and the hyperreals of non-standard analysis, and about how
Dedekind cuts and Cauchy sequences do not represent all of the
hyperreals, each of which is a real number.

There's still consideration of the powerset mapping result and
Cantor-Bernstein transitivity, it is a broad consideration that in my
view best fits within a foundationless set theory.

A variety of analytical results pertaining to EF are discussed, for
example with regards to Banach-Tarski.

Most considerations of transfinite cardinals have little or no
application in empirical or observable results. The ones that do have
alternatives.

Basically one way to consider the real numbers is that essentially
regardless of their field attributes, that there is a contiguous
sequence of them. There is a continuous sequence of them.

Regards,

Ross F. 

--
"Also, consider this:  the unit impulse function times 
one less twice the unit step function times plus/minus 
one is the mother of all wavelets."

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