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Topic: [HM] A question as to Cantor's Diagonal Method.
Replies: 13   Last Post: Oct 25, 2005 6:01 AM

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 Alexander Zenkin Posts: 50 Registered: 12/3/04
[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, ...},
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 1-1-correspondence between elements of the sets, X and N.
RAA-PROOF. Assume that |X| = |N|, BUT there is a rule/algorithm
producing a 1-1-correspondence 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