EDB — 23R

Zorn Lemma, Axiom of Choice, Zermelo’s Theorem[23R]

There are three fundamental statements in set theory, Zorn’s Lemma, the Axiom of Choice, and Zermelo’s Theorem. It is proven, within the Zermelo–Fraenkel axiomatics, that these are equivalent. See in Chap. 1 in [ 3 ] for an elementary presentation, based on the above defined theory. ¹

The first exercise presents some fundamental equivalent ways to state the Axiom of Choice.

E42

[02H]↺↻

E43

[2GF]↺↻

E43

[2BZ]↺↻

E43

[02D]↺↻

E43

[02M]↺↻

Many other exercise need Zorn’s Lemma, Axiom of Choice, Zermelo’s Theorem in their proof; to cite a few: [2BX]↺↻, [02S]↺↻, [04M]↺↻, [04P]↺↻, [04R]↺↻, [04Z]↺↻.