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The unpublished version given here says that given any set , there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 ac3 8859 for a more detailed
explanation. Theorem ac2 8858 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 ackm 8862
is slightly shorter
when the biconditional of ax-ac 8856 is expanded into implication and
negation. In axac3 8861 we allow the constant CHOICE
to represent the
Axiom of Choice; this simplifies the representation of theorems like
gchac 9076 (the Generalized Continuum Hypothesis implies
the Axiom of
Choice).
Standard textbook versions of AC are derived as ac8 8889,
ac5 8874, and
ac7 8870. The Axiom of Regularity ax-reg 8036 (among others) is used to
derive our version from the standard ones; this reverse derivation is
shown as theorem dfac2 8528. Equivalents to AC are the well-ordering
theorem weth 8892 and Zorn's lemma zorn 8904.
See ac4 8872 for comments about
stronger versions of AC.
In order to avoid uses of ax-reg 8036 for derivation of AC equivalents, we
provide ax-ac2 8860 (due to Kurt Maes), which is equivalent to
the standard
AC of textbooks. The derivation of ax-ac2 8860 from ax-ac 8856 is shown by
theorem axac2 8863, and the reverse derivation by axac 8864.
Therefore, new
proofs should normally use ax-ac2 8860 instead.
(New usage is discouraged.) (Contributed by NM,
18-Jul-1996.) |