
Re: Joel David Hamkins on definable real numbers in analysis
Posted:
Jun 25, 2013 5:27 PM


On Tuesday, June 25, 2013 1:43:36 PM UTC7, muec...@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 with the result that finished > > infinity as presumed for transfinite set theory is not a valid > > mathematical notion. >
The statement, "If n e N, then FIS_n(d) e S." does NOT imply that "d e S".
> > Regards, WM
ZG

