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