From: Norman D. Megill<ndm@shore.net>
Subject: Re: A question on Set Theory
Date: 16 Apr 2000 04:21:13 -0400
Newsgroups: sci.math.research
Summary: [missing]

In article <38F85A23.B5B13295@algebra.uni-linz.ac.at>,
Franz Binder  <Franz.Binder@algebra.uni-linz.ac.at> wrote:

[...]

>But there are more possibilietes that make sense (thus no canonical
>one), e.g.,
>1) A=B iff there is a bijection between them
>2) A=B iff there is an injection in both directions
>Note that Bernstein's lemma is not true in the constructive context
>(need Axiom of Choice).

Although "Bernstein's lemma" (Schroeder-Bernstein theorem?) has a
much shorter proof using AC, it does not need it and can be proved using
only the following axioms of ZF set theory:  Extensionality, Replacement,
Union, and Power.  Not even Infinity is needed.

--Norm