On 5 Feb., 17:06, fom <fomJ...@nyms.net> wrote: > 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.
Same is valid for the one-to-one correspondence of the natural numbers with the natural numbers.
> > In the context of the statements above, one cannot even put > Borel's accessible numbers in one-to-one correspondence with > the natural numbers.
That is correct because one cannot even put all natutal numbers in correspondence with all natural numbers.
For every line n of the list
1 - 1 2 - 1, 2 3 - 1, 2, 3 ...
we can state that there are not all natural numbers. Since there is never more than one line required to gather all numbers that appear in any two lines of finite index and since there are only lines with finite index, we can conclude that not all natural numbers are in the list.
They only could be in the list, if the existence of "actually infinitely many lines" would prevent to apply the proof by induction to the whole list. But that trick, mutilating logic, would also prevent to exclude the antidiagonal from the complete list. Why should logic remain valid in the latter case, if it has been toppled in the former?
"All" is different from "every", when "all" is understood in the matheological sense of more than all finite 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.
This correspondence is as impossible, as I have shown above, as finding a set of natural numbers with negative sum.