
Re: Joel David Hamkins on definable real numbers in analysis
Posted:
Jun 25, 2013 8:04 PM


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. > > Proof: There are infinitely many rationals q_k with this property. All > are in the list by definition. At most n of them are in the first n > lines of the list. Infinitely many must exist in the remaining part of > the list. So we have obtained: > > For all n exists k: > d_1, d_2, d_3, ..., d_n = q_k1, q_k2, q_k3, ..., q_kn. > This theorem it is not less important than Cantor's theorem: For all > k: d =/= q_k. > > Both theorems contradict each other
No.
> with the result that finished > infinity as presumed for transfinite set theory is not a valid > mathematical notion. > > Regards, WM
The result is that you are unbelievably stupid. A person of just normal stupidity would not proudly show off her stupidity every day.

