Date: Feb 5, 2013 11:06 AM
Author: fom
Subject: Re: Matheology § 210
On 2/5/2013 4:15 AM, WM wrote:

>

> Matheology § 210

>

> An accessible number, to Borel, is a number which can be described as

> a mathematical object. The problem is that we can only use some finite

> process to describe a real number so only such numbers are accessible.

> We can describe rationals easily enough, for example either as, say,

> one-seventh or by specifying the repeating decimal expansion 142857.

> Hence rationals are accessible. We can specify Liouville's

> transcendental number easily enough as having a 1 in place n! and 0

> elsewhere. Provided we have some finite way of specifying the n-th

> term in a Cauchy sequence of rationals we have a finite description of

> the resulting real number. However, as Borel pointed out, there are a

> countable number of such descriptions. Hence, as Chaitin writes: "Pick

> a real at random, and the probability is zero that it's accessible -

> the probability is zero that it will ever be accessible to us as an

> individual mathematical object."

> [J.J. O'Connor and E.F. Robertson: "The real numbers: Attempts to

> understand"]

> http://www-history.mcs.st-and.ac.uk/HistTopics/Real_numbers_3.html

>

> But how to pick this dark matter of numbers? Only accessible numbers

> can get picked. Unpickable numbers cannot appear anywhere, neither in

> mathematics nor in Cantor's lists. Therefore Cantor "proves" that the

> pickable numbers, for instance numbers that can appear as an

> antidiagonal of a defined list, i.e., the countable numbers, are

> uncountable.

It is important to state what Cantor's proof proves properly.

Cantor's proof proves that any assertion claiming to put the

real numbers in one-to-one correspondence with the natural

numbers is fallible.

In the context of the statements above, one cannot even put

Borel's accessible numbers in one-to-one correspondence with

the natural numbers.

"countable" and "uncountable" are labels that distinguish non-finite

parts of absolute infinity which cannot be put into one-to-one

correspondence with one another with "countable" referring

to any such part that can be put into one-to-one correspondence

with the natural numbers.

Since "the natural numbers" should be problematic here, it

might be best to add "by a rule".

But, then your fight is actually with Dirichlet, isn't it?