Alternative method An alternative method is the following. Let Φ(x) be the formula that says "x is inductive"; i.e. Φ(x)=(∅∈x∧∀y(y∈x→(y∪{y}∈x))). Informally, what we will do is take the intersection of all inductive sets. More formally, we wish to prove the existence of a unique set W such that ∀x(x∈W↔∀I(Φ(I)→x∈I)). (*) For existence, we will use the Axiom of Infinity combined with the Axiom schema of specification. Let I be an inductive set guaranteed by the Axiom of Infinity. Then we use the axiom schema of specification to define our set W={x∈I:∀J(Φ(J)→x∈J)} – i.e. W is the set of all elements of I, which also happen to be elements of every other inductive set. This clearly satisfies the hypothesis of (*), since if x∈W, then x is in every inductive set, and if x is in every inductive set, it is in particular in I, so it must also be in W. For uniqueness, first note that any set that satisfies (*) is itself inductive, since 0 is in all inductive sets, and if an element x is in all inductive sets, then by the inductive property so is its successor. Thus if there were another set W′ that satisfied (*) we would have that W′⊆W since W is inductive, and W⊆W′since W′is inductive. Thus W=W′. Let ω denote this unique element. This definition is convenient because the principle of induction immediately follows: If I⊆ω is inductive, then also ω⊆I, so that I=ω.■ (引用終り)
