On 21 Jun., 21:35, "Mike Terry" <news.dead.person.sto...@darjeeling.plus.com> wrote:
> > It is an irrational number. Add it at first position to the former > > list L0, obtain L1. Now construct the diagonal of L1 (according to a > > fixed scheme). It is certainly an irrational number. Add ist to the > > list L1 at first position, obtain list L2. Now construct the > > diagonal, ... and continue. I this way you get a countable set > > consisting of all rationals and of all diagonal numbers of these > > lists. > > No... What you get is an infinite sequence of lists (L0, L1, L2, ...) > > Each Ln in the sequence contains all rationals, and Ln contains the > antidiagonals for L0,...L(n-1). Ln doesn't contain its own antidiagonal, > and doesn't contain any antidiagonals for Lm with m>n.
Correct. Therefore the set of antidiagonals appears to be uncountable = unlistable. But it is countable. > > > This set is certainly not countable, because you can prove that > > there is always a diagonal number not in the list. > > What set?
The set of antidiagonals that can be constructed (by a given substitution rule) from an infinite sequence of lists.
> It seems you're thinking there is some "limit list" for the > sequence of lists (L0, L1, L2, ...), or do you mean a specific list Ln for > some n?
No, there is no limit list, there is an infinite sequences of lists. > > If you mean some kind of limit list, please define exactly what this is.
There is no limit. There is a set of numbers, namely all rational numbers and all antidiagonals. > > > On the other hand, > > the set is countable by construction. What now? > > That depends on how you answer my previous question!
Not at all. Every number that can be definied by any means, be it constructible, definable, computable, computable relative to a given list, X1, X2, X3 (where the Xn may be further notions to be devised by clever set theorists in order to avoid contradictions in set theory), every such number belongs to a countable set. That observation does not depend on any further definition.
By the way: Cantor proved (or thought to prove) that the set of definable reals is uncountable. Cantor did not believe in undefinable reals (as he wrote to Hilbert in August 1906). In fact an undefinable number is nonsense, because the basic property of a number is the trichotomy with other numbers.