Page 75 - Textos de Matemática Vol. 38

ORDINALS, COMPUTATIONS, AND MODELS OF SET THEORY 67
Proof . Note that the formula φ may contain further free parameters, which we do not mention for the sake of simplicity. Assume that ∃uφ(u). Take a point x = (x,Rx) such that φ(x). Since Rx is wellfounded one may take an Rx-minimal y ∈ dom(Rx) such that φ((y,Rx)). Letting y also denote the point (y, Rx) then φ(y). To prove the axiom, consider some point z ◭ y . Take vRxy such that z ≡ (v,Rx). By the Rx-minimal choice of y we have ¬φ((v,Rx)). Hence ¬φ(z). qed(7)
(8) The axiom of infinity holds in , i.e.,
∃x(∃yy ◭ x ∧ ∀y(y ◭ x → ∃z(z ◭ x ∧ ∀u(u ◭ z ↔ u ◭ y ∨ u ≡ y))))
Proof . In SO let ω be the smallest limit ordinal. We show that x = (ω, <↾ (ω + 1)2)
witnesses the axiom. Since (0, <↾ (ω + 1)2) ◭ (ω, <↾ (ω + 1)2) we have ∃yy ◭ x. Consider some y ◭ x. We may assume that y = (n,<↾ (ω+1)2) for some n < ω. Set
It is easy to check that
z = (n + 1, <↾ (ω + 1)2).
z ◭ x ∧ ∀u(u ◭ z ↔ u ◭ y ∨ u ≡ y).
Theorem 7.3. In the set theoretical universe V consider a class S ⊆ {x|x ⊆ Ord} such that S = (Ord,S,<,=,∈,G) is a model of the theory SO. Then there is a unique inner model (M,∈) of ZFC such that S = {v ∈ M|v ⊆ Ord}.
Proof. Define the model = (,≡,◭) from (Ord,S,<,=,∈,G) as above. Consider a point x = (x,Rx) ∈ . Then x is also an ordinal in the sense of V. In S, apply the recursion theorem to the wellfounded relation Rx and obtain an order-preserving map
σ : (dom(Rx), Rx) → (Ord, <). Transfer the map σ to V by defining
σ˜ = {(α,β)|S σ(α) = β} : dom(Rx) → Ord.
This map is order-preserving and witnesses that Rx is wellfounded in V . So (x, Rx) is a point in the sense of V . In V , define the interpretation function I : → V recursively by
Ix : dom(Rx) → V , by Ix(u) = {Ix(v)|v Rx u}, and I(x) = Ix(x).
qed(8)
✷