Generalizing Hartogs' Trichotomy Theorem

Details Generalizing Hartogs' Trichotomy Theorem Generalizing Hartogs' Trichotomy Theorem

2008-04-04

arxiv.org/abs/0804.0673v1

Mathematics

Logic

8 pages with an appendix by Andreas Blass

Scientific paper

A celebrated argument of F. Hartogs (1915) deduces the Axiom of Choice from
the hypothesis of comparability for any pair of cardinals. We show how each of
a sequence of seemingly much weaker hypotheses suffices. Fixing a finite number
$k>1$, the Axiom of Choice follows if merely any family of $k$ cardinals
contains at least one comparable pair.

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