On Oct 28, 2:56 am, WM <mueck...@rz.fh-augsburg.de> wrote: > On 28 Okt., 05:44, William Hughes <wpihug...@gmail.com> wrote: > > > > > > > > > > > > ? > > > You want to say here that aleph_0 is not larger than all natural > > > numbers? > > > Or do you want to say that 1/9 as decimal fractions has less than > > > aleph_0 digits? > > > no. "0.111... has more digits than all its finite approximations" > > has two possible meanings. > > > i. forall n in N, number of digits positions (0.111...) > > > number of digit positions(0.111...1 (n terms) ) > > > ii. number of digit positions (0.111...) > > > number of (union n in N (digit positions(0.111...1 (n > > terms) ) > > > i. is true and ii, is false. i. is trivial. To see ii. note that > > there is one finite approximation for each element of N. > > Each finite approximation adds one new digit position to the union. > > 0.111... has one digit for each element of N > > If 0.111... has not more digits than the union of all its finite > approximations, then we have three alternatives: > The Kolmogorov complexity of infinite strings is not larger than all > finite complexities (cp. Das Kalenderblatt 111028, text in English)http://www.hs-augsburg.de/~mueckenh/KB/KB%20801-.pdf > or the inequality of Kraft is wrong (in fact it is correct) > (Das Kalenderblatt 111025)http://www.hs-augsburg.de/~mueckenh/KB/KB%20801-.pdf > or the infinite paths of the binary tree are nothing but unions of all > finite paths and as such form a countable set. >
A pretty typical WM response. Given a simple argument he does not address it or his original argument. Instead he gives three new complicated arguments.
However, references to Kolmogorov complexity and the Kraft inequality will not change the fact that there is one line for every element of N (and no other lines) and one column for every element of N (and no other columns).
