Choice, axiom of
Choice, axiom of
or Zermelo’s axiom, is the name given to an assumption of logical or logico-mathematical character which may be stated as followsGiven a class K whose members are non-empty classes, there exists a (one-valued) monadic function f whose range is K, such that f(x) &isinx for all members x of K. This had often been employed unconsciously or tacitly by mathematicians — and is apparently necessary for the proofs of certain important mathematical theorems — but was first made explicit by Zermelo in 1904, who used it in a proof that every class can be well-ordered. Once explicitly stated the assumption was attacked by many mathematicians as lacking in validity or as not of legitimately mathematical character, but was defended by others, including Zermelo.
An equivalent assumption, called by Russell the multiplicative axiom and afterwards adopted by Zermelo as a statement of his Auswahl-prinzip, is as followsGiven a class K whose members are non-empty classes no two of which have a member in common, there exists a class A (the Auswahlmenge) all of whose members are members of members of K and which has one and only one member in common with each member of K. Proof of equivalence of the multiplicative axiom to the axiom of choice is due to Zermelo. — A.C.
E. Zermelo,
Beweis, dass jede Menge wohlgeord net werden kann, Mathematische Annalen, vol. 59 (1904), pp. 514-516.
B. Russell,
On some difficulties in the theory of transfinite numbers and order types. Proceedings of the London Mathematical Society, ser 2. vol. 4 (1906), pp 29-53.
E. Zermelo,
Neuer Beweis fur die Moglichkeit einer Wohlordnung, Mathematische Annalen, vol. 65 (1908), pp. 107-128.
K. Gdel,
The Consistency of the Axiom of Choice and of the Generalized Continuum Hypothesis with the Axioms of Set Theory. Princeton, N.J., 1940.