Virgil
Posts:
8,833
Registered:
1/6/11


Re: Joel David Hamkins on definable real numbers in analysis
Posted:
Jun 25, 2013 5:09 PM


In article <c943131fa05448f486cb38c7c3375948@googlegroups.com>, mueckenh@rz.fhaugsburg.de wrote:
> On Tuesday, 25 June 2013 17:07:36 UTC+2, dull...@sprynet.com wrote: > >> There are not uncountably many reals, neither in the natural order nor in > >> the wellorder. > > > > That's very funny. > > That's easily provable: > > Consider a Cantorlist that contains a complete sequence (q_k) of all > rational numbers q_k. The first n digits of the antidiagonal d are > d_1, d_2, d_3, ..., d_n. It can be shown *for every n* that the Cantor > list beyond line n contains infinitely many rational numbers q_k that > have the same sequence of first n digits as the antidiagonal d.
Thus at most proving that one countable set of rationals surjects onto another countable set of rationals, but does no more than that. >
> For all n exists k: > d_1, d_2, d_3, ..., d_n = q_k1, q_k2, q_k3, ..., q_kn.
So what? That does not prove that there exists any k for all n.
Quantifier dyslexia cripples WM's arguments again!
WM's claims above do NOT establish existence of any k for which d_n = q_k_n for ALL n in N, which is what WM would need to prove his false claim. 

