Definition:Cantor's Theory of Sets

Definition

Georg Cantor's set theory builds upon Richard Dedekind's notion that an infinite set can be placed in one-to-one correspondence with a proper subset of itself.

However, he noticed that not all infinite sets are of the same cardinality.

While he appreciated that the sets of integers, rational numbers and algebraic numbers have the same cardinality as each other, aleph-null ($\aleph_0$), the set of real numbers have a different (larger) cardinality.

On noticing that the power set of a set has a strictly larger cardinality than the set itself, he realised that there is an infinite number of these transfinite numbers.

Also see

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Source of Name

This entry was named for Georg Cantor.

Historical Note

Georg Cantor developed his theory of sets from $1874$.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Cantor's theory of sets
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Cantor's theory of sets