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


Re: Joel David Hamkins on definable real numbers in analysis
Posted:
Jun 26, 2013 5:53 PM


In article <u43ms8do4814dgi1l1b0fh79ju5r42teu3@4ax.com>, dullrich@sprynet.com wrote:
> 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
WM's quantifier dyslexia strikes again! WM will never be able to make it as a mathematician until he cures his quantifier dyslexia.
And also escape from the other corruptions of his wild weird world of WMytheology 

