In article <email@example.com>, WM <firstname.lastname@example.org> wrote:
> 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.
What is "fallible" about "for all n in |N, n <--> n" ?
Unless one is tied up in WMytheology, nothing is wrong with it. > > > > > 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
What is "fallible" about "for all n in |N, n <--> n" ? > ... > > we can state that there are not all natural numbers.
You can state that the moon is made of green cheese, but stating something does not make it true.
> 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.
Show us that there is a natural number missing from "for all n in |N, n <--> n"
> > "All" is different from "every", when "all" is understood in the > matheological sense of more than all finite numbers.
Only in WMytheology. > > > > "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
That WM has claimed something does not mean that he has shown it. and usually seems to mean he has not SHOWN it at all. --