Description: Axiom of Choice using
abbreviations. The logical equivalence to ax-ac 9618
can be established by chaining aceq0 9276 and aceq2 9277. A standard
textbook version of AC is derived from this one in dfac2a 9287, and this
version of AC is derived from the textbook version in dfac2b 9288, showing
their logical equivalence (see dfac2 9289).
The following sketch will help you understand this version of the axiom.
Given any set 𝑥, the axiom says that there exists a
𝑦
that is a
collection of unordered pairs, one pair for each nonempty member of
𝑥. One entry in the pair is the member
of 𝑥, and the other
entry is some arbitrary member of that member of 𝑥. Using the
Axiom of Regularity, we can show that 𝑦 is really a set of
ordered
pairs, very similar to the ordered pair construction opthreg 8812. The
key theorem for this (used in the proof of dfac2b 9288) is preleq 8810.
With this modified definition of ordered pair, it can be seen that 𝑦
is actually a choice function on the members of 𝑥.
For example, suppose 𝑥 = {{1, 2}, {1, 3}, {2, 3, 4}}.
Let us try 𝑦 = {{{1, 2}, 1}, {{1, 3}, 1},
{{2, 3, 4}, 2}}. For the member (of 𝑥)
𝑧 =
{1, 2}, the only assignment to 𝑤 and 𝑣 that
satisfies the axiom is 𝑤 = 1 and 𝑣 = {{1, 2}, 1}, so
there is exactly one 𝑤 as required. We verify the other two
members
of 𝑥 similarly. Thus, 𝑦
satisfies the axiom. Using our
modified ordered pair definition, we can say that 𝑦 corresponds to
the choice function {⟨{1, 2}, 1⟩, ⟨{1,
3}, 1⟩,
⟨{2, 3, 4}, 2⟩}. Of course other
choices for 𝑦 will
also satisfy the axiom, for example
𝑦 =
{{{1, 2}, 2}, {{1, 3}, 1},
{{2, 3, 4}, 4}}. What AC tells us is that there
exists at
least one such 𝑦, but it doesn't tell us which one.
(New usage is discouraged.) (Contributed by NM,
19-Jul-1996.) |