On 4/12/2013 11:41 AM, WM 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 > No statement about n > k is appropriate or possible from the facts. >
The facts are that unless WM assumes that an endless application of long division by the Euclidean algorithm is constant WM has no "knowledge", in the sense of his finitistic claims, upon which to base his denial.
WM has not "proved" that each truncated string is, in fact, the representation which he is claiming it to be with an abbreviation.
When WM writes,
"No statement about n > k is appropriate or possible from the facts."
he, in fact, is making a statement about n > k.
Anything that might confuse the facts is appropriate and possible from WM.
>> Cantor negated : >> >> 3) exists k , forall n , d_n == q_kn > > No. That negation is valid only for all n =< k. >
No. The only invalid statements here are WM's
>> 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
In WM's world, procreation seems to involve immaculate conceptions.
>> >> 2) forall children , exists woman , woman is child's ancestor >> 3) exists woman , forall children , woman is child's ancestor.
Oddly, based on mitochondrial DNA, genetic analyses have made exactly this claim.