**Description: **Axiom of Choice using
abbreviations. The logical equivalence to ax-ac 8303
can be established by chaining aceq0 7963 and aceq2 7964. A standard
textbook version of AC is derived from this one in dfac2a 7974, and this
version of AC is derived from the textbook version in dfac2 7975.
The following sketch will help you understand this version of the
axiom. Given any set , the axiom says that there exists a
that is a collection of unordered pairs, one pair for each non-empty
member of . One
entry in the pair is the member of , and
the other entry is some arbitrary member of that member of .
Using the Axiom of Regularity, we can show that is really a set of
*ordered* pairs, very similar to the ordered pair construction
opthreg 7537. The key theorem for this (used in the proof
of dfac2 7975) is
preleq 7536. With this modified definition of ordered
pair, it can be
seen that is
actually a choice function on the members of .
For example, suppose
. Let us try
. For the member (of )
, the only assignment to and that
satisfies the axiom is and , so
there is exactly one as required. We verify the other two members
of similarly.
Thus, satisfies the
axiom. Using our
modified ordered pair definition, we can say that corresponds to
the choice function
. Of course other choices for will
also satisfy the axiom, for example
. What AC tells us is that there exists at
least one such ,
but it doesn't tell us which one.
(New usage is discouraged.) (Contributed by NM,
19-Jul-1996.) |