
Re: Joel David Hamkins on definable real numbers in analysis
Posted:
Jun 26, 2013 11:48 AM


On Tue, 25 Jun 2013 13:43:36 0700 (PDT), 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
Erm, no they don't. If your "for every n there exists k such that" was instead "there exists k such that for every n" then they would contradict each other.
Don't feel bad about this. Confusing the order of quantifiers that way is very common among people who haven't learned to reason carefully.
>with the result that finished >infinity as presumed for transfinite set theory is not a valid >mathematical notion. > >Regards, WM

