Schroeder–Bernstein theorem

2020 Mathematics Subject Classification: Primary: 03E20 [MSN][ZBL]

Cantor–Bernstein theorem

For sets $A$ and $B$, if there are injections from $A$ to $B$ and from $B$ to $A$ (equivalently, each is equipotent to a subset of the other), then there is a bijection between $A$ and $B$ (they are equipotent sets).

In cardinal arithmetic, if we let $\mathfrak{a} \le \mathfrak{b}$ denote the property that some set of cardinality $\mathfrak{a}$ has an injection to a set of cardinality $\mathfrak{b}$, then $\mathfrak{a} \le \mathfrak{b}$ and $\mathfrak{b} \le \mathfrak{a}$ implies $\mathfrak{a} = \mathfrak{b}$.

The theorem was conjectured by Georg Cantor by 1895 and proved by Felix Bernstein in 1897. Dedekind obtained a further proof in 1897. Schroeder's proof of 1898 was found to be flawed by 1902.

References

• P. R. Halmos, "Naive Set Theory", Springer (1960) ISBN 0-387-90092-6
• Michael Potter, "Set Theory and its Philosophy : A Critical Introduction", Oxford University Press (2004) ISBN 0-19-155643-2