Tilted Forum Project Discussion Community - View Single Post - Cantor-Schroder-Bernstein Theorem

dimbulb · 10-21-2003, 05:18 PM

http://mathworld.wolfram.com/CardinalComparison.html

Quote:

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.