
Re: Joel David Hamkins on definable real numbers in analysis
Posted:
Jun 26, 2013 12:01 PM


On Wednesday, 26 June 2013 17:48:25 UTC+2, dull...@sprynet.com wrote: > > >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.
That is the old nonsense with men and women or forks and knifes. In a linear set there is no chance to apply the quantifier trick. It leads to mess. Even you should understand this, if you tried.
Consider the sequence
0.1 0.11 0.111 ...
There is no term 1/9. But the diagonal is assumed to be 1/9.
All FIS of 1/9 are in the lines. 1/9 is not a line. But 1/9 is note elewhere either, because it cannot contain more numbers than are in FIS.
The complete symmetry of the triangle of digits 1 can be shown by constructing it in a symmetrical way:
a
a bb
c ac bbc
d dc dac dbbc
and so on. There is no side with aleph_0 symbols unless all sides have aleph_0 symbols in ther limit.
Therefore For all n exists k: >d_1, d_2, d_3, ..., d_n = q_k1, q_k2, q_k3, ..., q_kn. means the same as with quantifiers reversed (simply because there is no "for all").
Regards, WM

