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純粋・応用数学・数学隣接分野（含むガロア理論）20

PC,ｽﾏﾎ,PHSは ULA べっかんこ公式(ｽﾏﾎ) 公式(PC) で書き込んでください｡

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引用切替：ﾚｽｱﾝｶｰのみ

>>631
> つづき
> 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)
> 以上

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