abstract
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Comparing cardinalities in Zermelo's system

Lorenz Halbeisen

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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.