Cardinal Comparison

For any sets A and B, their cardinal numbers satisfy |A|<=|B| iff there is a one-to-one function f from A into B (Rubin 1967, p. 266; Suppes 1972, pp. 94 and 116). It is easy to show this satisfies the reflexive and transitive axioms of a partial order. However, it is difficult to show the antisymmetry property, whose proof is known as the Schröder-Bernstein theorem. To show the trichotomy property, one must use the axiom of choice.

Although an order type can be defined similarly, it does not seem usual to do so.

See also

Schröder-Bernstein Theorem

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Rubin, J. E. Set Theory for the Mathematician. New York: Holden-Day, 1967.Suppes, P. Axiomatic Set Theory. New York: Dover, 1972.

Referenced on Wolfram|Alpha

Cardinal Comparison

Cite this as:

Weisstein, Eric W. "Cardinal Comparison." From MathWorld--A Wolfram Web Resource.

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