THOUGHTS ON THE CANTOR-BERNSTEIN THEOREM
作者:
B. Banaschewski,
G.C.L. Brummer,
期刊:
Quaestiones Mathematicae
(Taylor Available online 1986)
卷期:
Volume 9,
issue 1-4
页码: 1-27
ISSN:1607-3606
年代: 1986
DOI:10.1080/16073606.1986.9632106
出版商: Taylor & Francis Group
关键词: Primary 18825;secondary 01A55;03E25;04A05;06E99
数据来源: Taylor
摘要:
The usual proofs of the well-known set-theoretical theorem “Given one-one maps f: A → B and g:B → A, there exists a one-one onto map h:A → B” actually produce a map h:A → B contained in the relation f U g−1. Considering Tarski's Fixpoint Theorem as the implicit basic ingredient of such proofs. We examine several classical proofs/starting with Dedekind (1887), and illuminate their common feature by means of the categorical notion of a natural fixpoint. We consider a categorical form (CBT) of the theorem (with h ⊆ f Ug−1) in a variety of contexts, obtaining some examples of categories where CBT holds and others where it fails. Among other results we prove for a toposE, (1) CBT holds ifEis Boolean, and conversely ifEhas a natural number object; (2) The Axiom of Choice inEimplies a dual version of CBTI and conversely ifEhas splitting supports and a natural number object.
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