Comparing cardinalities in Zermelo's system

The aim of this note is to define and compare cardinalities in Zermelo's system of 1908 (without using the Axiom of Choice). The main tool to compare cardinalities in Set Theory (with or without the Axiom of Choice) is the Cantor-Bernstein Theorem. We shall present two different proofs of the Cantor-Bernstein Theorem and shall give some historical background. In particular, we shall see that this theorem was proved independently by Cantor, Dedekind, Bernstein, Korselt, Zermelo, and Peano, but not by Schröder - even though it is sometimes cited as Schröder-Bernstein Theorem.

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