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


On Wednesday, 26 June 2013 02:04:40 UTC+2, Ralf Bader 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.
No.
If you do not agree, please find any digit of the diagonal which, together with all its predecessors, is not in a line of the list. Of course you are unable to do so. Therefore your bigmouthed "no" is of the same value as your usual posts.
In mathematics we have: If every digits of d together with all its predecessors is in a set of numbers, then d is in the set of numbers. Since you cannot distinguish it by a digit.
Your simple mistake is to believe, that a number can be defined by infinitely many digits that is wrong. For example the number 1/9 is not defined by aleph_0 digits. 1/9 is the limit of the sequence
0.1 0.11 0.111 ...
but it is not definable by infinitely many digits. Only matheologians believe that indefinitely many digits could be written or read or could be used in any way. 1/9 is the limit  but it is neither in a line nor in a columns nor in the diagonal. Therefore my theorem is of the same value as Cantor's.
Regards, WM

