In article <email@example.com>, WM <firstname.lastname@example.org> wrote:
> 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
But does not ever require that there exist any n's less than k for which d_n =/= q_kn, so does not say any more than that d_k =/= q_kk.
> 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.
Not even for all n < k, only for 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.
We can certainly have 2, at least back tos Eve, but not 3, as no woman can be her own ancestor.
Thus we have another case of quantifier dyslexia. --