```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 thereal numbers in one-to-one correspondence with the naturalnumbers is fallible.In the context of the statements above, one cannot even putBorel's accessible numbers in one-to-one correspondence withthe natural numbers."countable" and "uncountable" are labels that distinguish non-finiteparts of absolute infinity which cannot be put into one-to-onecorrespondence with one another with "countable" referringto any such part that can be put into one-to-one correspondencewith the natural numbers.Since "the natural numbers" should be problematic here, itmight be best to add "by a rule".But, then your fight is actually with Dirichlet, isn't it?
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