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The unpublished version given here says that given any set , there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 8743 for a more detailed
explanation. Theorem ac2 8742 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 8746
is slightly shorter
when the biconditional of ax-ac 8740 is expanded into implication and
negation. In axac3 8745 we allow the constant CHOICE
to represent the
Axiom of Choice; this simplifies the representation of theorems like
gchac 8960 (the Generalized Continuum Hypothesis implies
the Axiom of
Choice).
Standard textbook versions of AC are derived as ac8 8773,
ac5 8758, and
ac7 8754. The Axiom of Regularity ax-reg 7919 (among others) is used to
derive our version from the standard ones; this reverse derivation is
shown as theorem dfac2 8412. Equivalents to AC are the well-ordering
theorem weth 8776 and Zorn's lemma zorn 8788.
See ac4 8756 for comments about
stronger versions of AC.
In order to avoid uses of ax-reg 7919 for derivation of AC equivalents, we
provide ax-ac2 8744 (due to Kurt Maes), which is equivalent to
the standard
AC of textbooks. The derivation of ax-ac2 8744 from ax-ac 8740 is shown by
theorem axac2 8747, and the reverse derivation by axac 8748.
Therefore, new
proofs should normally use ax-ac2 8744 instead.
(New usage is discouraged.) (Contributed by NM,
18-Jul-1996.) |