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2010•arXiv (Cornell University)Open access

Cantor versus Cantor

Antonio León

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Abstract

This paper examines the possibilities of extending Cantor's two arguments on the uncountable nature of the set of real numbers to one of its proper denumerable subsets: the set of rational numbers. The paper proves that, unless certain restrictive conditions are satisfied, both extensions are possible. It is therefore indispensable to prove that those conditions are in fact satisfied in Cantor's theory of transfinite sets. Otherwise that theory would be inconsistent.

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This paper examines the possibilities of extending Cantor's two arguments on the uncountable nature of the set of real numbers to one of its proper denumerable subsets: the set of rational numbers. The paper proves that, unless certain restrictive conditions are satisfied, both extensions are possible. It is therefore indispensable to prove that those conditions are in fact satisfied in Cantor's theory of transfinite sets. Otherwise that theory would be inconsistent.

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Available abstract

This paper examines the possibilities of extending Cantor's two arguments on the uncountable nature of the set of real numbers to one of its proper denumerable subsets: the set of rational numbers. The paper proves that, unless certain restrictive conditions are satisfied, both extensions are possible. It is therefore indispensable to prove that those conditions are in fact satisfied in Cantor's theory of transfinite sets. Otherwise that theory would be inconsistent.

Key concepts: Uncountable set, Transfinite number, Cantor's diagonal argument, Countable set, Cantor set, Cantor function, Mathematics, Cardinal number (linguistics)

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