View source for Schroeder–Bernstein theorem

{{MSC|03E20}}

''Cantor–Bernstein theorem''

For sets $A$ and $B$, if there are [[injection]]s 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}}

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