On 12 Apr., 17:24, Dan <dan.ms.ch...@gmail.com> wrote:
> They do not contradict each other : > Cantor's affirmation (in its full form) is : > > 1) forall k , exist n , d_n =/= q_kn
Tricky! No, please be careful. Cantor shows exactly: forall k: d_k =/= q_kk Not more and not less.
This can be extended to forall k, exists n =< k: d_n =/= q_kn No statement about n > k is appropriate or possible from the facts.
> Cantor negated : > > 3) exists k , forall n , d_n == q_kn
No. That negation is valid only for all n =< k.
> swapping the order of quantifiers has important consequences .
That depends on the structure of the set. In linearly ordered sets like chains of mother-child we have > > 2) forall children , exists woman , woman is child's ancestor > 3) exists woman , forall children , woman is child's ancestor.