On Apr 12, 5:27 pm, WM <mueck...@rz.fh-augsburg.de> wrote: > On 12 Apr., 16:14, dullr...@sprynet.com wrote: > > > >Cantor's list contains real numbers r as binary or decimal fractions. > > >Real numbers, however, are /limits/ of binary or decimal fractions. > > > No, it _is_ a list of limits of decimal fractions. > > There is no such list possible unless you give finite definitions. > It is impossible to define a number by writing an iinfinite sequence. > > > > > It passes belief that a person could actually > > think that such a well known proof could contain > > such a simple error, without that error being > > noticed by any of the thousands of mathematicians > > who've studied the subject for a century or so. > > This assumption may be the reason that Cantor's "proof" has been > believed over 100 years. Try to understand the following. (If you are > not a too slow thinker, this will happen before midnight.) > > Consider a Cantor-list that contains a complete sequence (q_k) of all > rational numbers q_k. The first n digits of the anti-diagonal d are > d_1, d_2, d_3, ..., d_n. It can be shown *for every n* that the > Cantor- > list beyond line n contains infinitely many rational numbers q_k that > have the same sequence of first n digits as the anti-diagonal d. > > Proof: There are infinitely many rationals q_k with this property. > All > are in the list by definition. At most n of them are in the first n > lines of the list. Infinitely many must exist in the remaining part > of > the list. So we have obtained: > > 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 with the result that finished > infinity as presumed for transfinite set theory is not a valid > mathematical notion. > > Regards, WM
They do not contradict each other : Cantor's affirmation (in its full form) is :
1) forall k , exist n , d_n =/= q_kn
Your (valid) theorem is :
2) forall n , exists k , d_1 = q_k1 and d_2 = q_k2 and ..... d_n = q_kn
The negation of Cantor theorem would reverse the quantifiers , that is : Cantor negated :
3) exists k , forall n , d_n == q_kn
Now , this sounds similar, but not the same as your theorem . Let's put side by side a simplified version of your theorem , and the Cantor negation:
2) forall n , exists k d_n = q_kn //your theorem 3) exists k , forall n , d_n == q_kn //Cantor's negative
They look remarkably the same, but they say different things , swapping the order of quantifiers has important consequences .
2) forall children , exists woman , woman is child's mother
//this has the same structure as your theorem . It says "every child has a woman such that the woman is it's mother"
3) exists woman , forall children , woman is child's mother
//this has the same structure as the Cantor negation . It says "there exists a (single , unique ) woman, who is the mother of every child" .