Axiom of Choice

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axiom of choice

[¦ak·sē·əm əv ′chȯis]
The axiom that for any family A of sets there is a function that assigns to each set S of the family A a member of S.

Axiom of Choice

(AC, or "Choice") An axiom of set theory:

If X is a set of sets, and S is the union of all the elements of X, then there exists a function f:X -> S such that for all non-empty x in X, f(x) is an element of x.

In other words, we can always choose an element from each set in a set of sets, simultaneously.

Function f is a "choice function" for X - for each x in X, it chooses an element of x.

Most people's reaction to AC is: "But of course that's true! From each set, just take the element that's biggest, stupidest, closest to the North Pole, or whatever". Indeed, for any finite set of sets, we can simply consider each set in turn and pick an arbitrary element in some such way. We can also construct a choice function for most simple infinite sets of sets if they are generated in some regular way. However, there are some infinite sets for which the construction or specification of such a choice function would never end because we would have to consider an infinite number of separate cases.

For example, if we express the real number line R as the union of many "copies" of the rational numbers, Q, namely Q, Q+a, Q+b, and infinitely (in fact uncountably) many more, where a, b, etc. are irrational numbers no two of which differ by a rational, and

Q+a == q+a : q in Q

we cannot pick an element of each of these "copies" without AC.

An example of the use of AC is the theorem which states that the countable union of countable sets is countable. I.e. if X is countable and every element of X is countable (including the possibility that they're finite), then the sumset of X is countable. AC is required for this to be true in general.

Even if one accepts the axiom, it doesn't tell you how to construct a choice function, only that one exists. Most mathematicians are quite happy to use AC if they need it, but those who are careful will, at least, draw attention to the fact that they have used it. There is something a little odd about Choice, and it has some alarming consequences, so results which actually "need" it are somehow a bit suspicious, e.g. the Banach-Tarski paradox. On the other side, consider Russell's Attic.

AC is not a theorem of Zermelo Fr?nkel set theory (ZF). G?del and Paul Cohen proved that AC is independent of ZF, i.e. if ZF is consistent, then so are ZFC (ZF with AC) and ZF(~C) (ZF with the negation of AC). This means that we cannot use ZF to prove or disprove AC.
References in periodicals archive ?
21) For the controversial status of the axiom of choice among other axioms of set theory, see Yiannis Moschovakis, Notes on Set Theory (New York: Springer, 2006).
Concerning the theorems mentioned above, the Axiom of Choice turns out to be indeed necessary: In section 4, we construct transitive models of Zermelo-Fraenkel set theory without the Axiom of Choice (ZF) containing a real closed field K, but no integer part of K.
We briefly summarize some notions from set theory that are necessary to understand the central tool for constructing a choice-free universe with an RCF without an integer part, namely Lemma 3 below, originally used by Hodges to show the dependence of several algebraic constructions on the Axiom of Choice ([H1]).
Intuitionism grew out of objections to results that arose in part from the axiom of choice.
14) On the other hand, John Byl, who opts for mathematical realism, attempts to ground a portion of mathematics--including the law of noncontradiction, the axiom of choice, and notions of a completed infinity--on attributes of God found in the scriptures.
Moreover, something like an Axiom of Choice, certainly its existential quality, is behind the controversial ontology to which Lewis refers in On the Plurality of Worlds.
But, thanks to the Axiom of Choice, we don't actually have to undertake this impossible task.
Specifically it is the Axiom of Choice that elevates the human animal to the level of a potential subject.
Hintikka holds that the relevant version of the axiom of choice is (or should be) unproblematic in constructive contexts.
Anyone seriously interested in the logical structure of our knowledge of identity, the concept of discriminability, vagueness and sorites paradoxes, philosophical applications of the Axiom of Choice, and the logical status and function of identity criteria will find that this book amply repays the careful study which it demands.
Indeed, since the Axiom of Choice fails in some models of ZFA, and V=L implies AC, we know that some models of ZFA contain nonconstructible sets.