>>578 つづき 2. The Background to Zermelo's Axiomatisation 2.1 Hilbert's Axiomatic Method
2.1.2 Proof analysis and Zermelo's Well-Ordering Theorem [WOT]
2.2 The Well-Ordering Problem and the Well-Ordering Theorem 2.2.1 The importance of the problem before Zermelo
2.2.5 The Axioms of the 1908 WOT Paper He also adds the Axiom of Infinity, to guarantee that there are infinite sets, and the Axiom of Extensionality, which codifies the assumption that sets are really determined by their members, and not by the accidental way in which these members are selected. In addition, as we have noted, he now calls the Axiom of Choice by this name.
3.2.3 Cardinality It was pointed out by both Fraenkel and Skolem in the early 1920s that Zermelo's theory cannot provide an adequate account of cardinality. The axiom of infinity and the power set axiom together allow the creation of sets of cardinality ≧ ｱﾚﾌn for each natural number n, but this (in the absence of a result showing that 2^ｱﾚﾌ0 > ｱﾚﾌn for every natural number n) is not enough to guarantee a set whose power is ≧ ｱﾚﾌω, and a set of power ｱﾚﾌω is a natural next step (in the Cantorian theory) after those of power ｱﾚﾌn. Fraenkel proposed a remedy to this (as did Skolem independently) by proposing what was called the Ersetzungsaxiom, the Axiom of Replacement (see Fraenkel 1922: 231 and Skolem 1923: 225?226). This says, roughly, that the ‘functional image’ of a set must itself be a set, thus if a is a set, then {F(x) : x ∈ a} must also be a set, where ‘F’ represents a functional correspondence. Such an axiom is certainly sufficient; assume that a0 is the set of natural numbers {0, 1, 2, …}, and now assume that to each number n is associated an an with power ｱﾚﾌn. Then according to the replacement axiom, a = {a0, a1, a2, …} must be a set, too. This set is countable, of course, but (assuming that the an are all disjoint) the union set of a must have cardinality at least ｱﾚﾌω. つづく
