Description: Axiom of Choice. The
Axiom of Choice (AC) is usually considered an
extension of ZF set theory rather than a proper part of it. It is
sometimes considered philosophically controversial because it asserts
the existence of a set without telling us what the set is. ZF set
theory that includes AC is called ZFC.
The unpublished version given here says that given any set , there
exists a that is a collection of
unordered pairs, one pair for
each nonempty member of . One entry in the pair is
the member of
, and the other entry is some arbitrary member of
that member of
. See the rewritten version ac38863
for a more detailed
explanation. Theorem ac28862 shows an equivalent written compactly with
restricted quantifiers.
This version was specifically crafted to be short when expanded to
primitives. Kurt Maes' 5-quantifier version ackm8866
is slightly shorter
when the biconditional of ax-ac8860 is expanded into implication and
negation. In axac38865 we allow the constant to represent the
Axiom of Choice; this simplifies the representation of theorems like
gchac9080 (the Generalized Continuum Hypothesis implies
the Axiom of
Choice).
Standard textbook versions of AC are derived as ac88893,
ac58878, and
ac78874. The Axiom of Regularity ax-reg8039 (among others) is used to
derive our version from the standard ones; this reverse derivation is
shown as theorem dfac28532. Equivalents to AC are the well-ordering
theorem weth8896 and Zorn's lemma zorn8908.
See ac48876 for comments about
stronger versions of AC.
In order to avoid uses of ax-reg8039 for derivation of AC equivalents, we
provide ax-ac28864 (due to Kurt Maes), which is equivalent to
the standard
AC of textbooks. The derivation of ax-ac28864 from ax-ac8860 is shown by
theorem axac28867, and the reverse derivation by axac8868.
Therefore, new
proofs should normally use ax-ac28864 instead.
(New usage is discouraged.) (Contributed by NM,
18-Jul-1996.)