
[HM] A question as to Cantor's Diagonal Method.
Posted:
Aug 18, 2005 4:54 AM


Dear All,
Whether the following fact is known/described anywhere in mathematical literature?
Here X=[0,1], N={1,2,3, ...}, RAA = Reductio ad Absurdum, CDM = Cantor's Diagonal Method. Z  a cardinality of a set Z for any Z.
THEOREM 5. If X = N, then there is NOT a rule/algorithm producing a 11correspondence between elements of the sets, X and N. RAAPROOF. Assume that X = N, BUT there is a rule/algorithm producing a 11correspondence between elements of X and N. Then the rule/algorithm generates a list, x1, x2, x3, . . . , (1) containing ALL real numbers from X. Applying CDM to the list (1), Cantor generates a new real number, which doesn't belong to the list (1). So, the given list (1) contains NOT ALL real numbers from X. Contradiction. Q.E.D.
Alexander Zenkin

