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?