axiom of choice
The axiom of choice is an axiom in set theory that is one of the most controversial axioms in mathematics; it was formulated in 1904 by the German mathematician Ernst Zermelo (1871–1953) and, at first, seems obvious and trivial. Imagine there are many – possibly an unlimited number of – boxes in front of you, each of which has at least one thing in it. The axiom of choice (AC) says simply that you can always choose one item out of each box. More formally, if S is a collection of non-empty sets, then there exists a set that has exactly one element in common with every set S of S. Put another way, there exists a function f with the property that, for each set S in the collection, f(S) is a member of S. Bertrand Russell summed it up neatly: "To choose one sock from each of infinitely many pairs of socks requires the Axiom of Choice, but for shoes the Axiom is not needed. His point is that the two socks in a pair are identical in appearance, so, to pick one of them, we have to make an arbitrary choice. For shoes, we can use an explicit rule, such as "always choose the left shoe." Russell specifically mentions infinitely many pairs, because if the number is finite then AC is superfluous: we can pick one member of each pair using the definition of "nonempty" and then repeat the operation finitely many times using the rules of formal logic.
AC lies at the heart of a number of important mathematical arguments and results. For example, it is equivalent to the well-ordering principle, to the statement that for any two cardinal numbers m and n, then m < n or m = n or m > n, and to Tychonoff's theorem (the product of any collection of compact spaces in topology is compact). Other results hinge upon it, such as the assertion that every infinite set has a denumerable subset. Yet AC was strongly attacked when it was first suggested, and still makes some mathematicians uneasy. The central issue is what it means to choose something from the sets in question and what it means for the choosing function to exist. This problem is brought into sharp focus when S happens to be the collection of all nonempty subsets of the real numbers. No one has ever found a suitable choosing function for this collection, and there are good reasons to suspect that no one ever will. AC just mandates that there is such function. Because AC conjures up sets without offering workable procedures, it is said to be nonconstructive, as are any theorems whose proofs involve AC. Another reason that some mathematicians aren't greatly enamored with AC is that it implies the existence of some bizarre counter-intuitive objects, the most famous and notorious example of which is the Banach-Tarski paradox. The main reason for accepting AC, as the majority of mathematicians do (albeit often reluctantly), is that it is useful. However, as a result of work by Kurt Gödel and, later, by Paul Cohen, it has been proven to be independent of the remaining axioms of set theory. Thus there are no contradictions in choosing to reject it; among the alternatives are to adopt a contradictory axiom or to use a completely different framework for mathematics, such as category theory.