Recursion Theory

It is immediate that J0 = ∈ M. If δ > 0, then π−1(δ)< β. Then 〈Jβ, ∈〉 ⊨ (∃x) (∃s)φ(x, π−1(δ), s) and so 〈M, ∈〉 ⊨ (∃x)(∃s)φ(x, δ, s). Let x ∈ M and let s ∈ M such that 〈M, ∈〉 ⊨ φ(x, δ, s). By the absoluteness of φ(x, δ, s), any admissible structure containing x, s and δ satisfies φ(x, δ, s). By Proposition 3.3.3, Jδ = x ∈ M.

Furthermore, for any x ∈ M, there exists a δ < β such that π−1(x)∈ Jδ. Hence 〈Jβ, ∈〉 ⊨ (∃δ)(∃y)(∃s)(π−1(x)∈ y ∧ φ(y, δ, s)). Then 〈M, ∈〉 ⊨ (∃δ)(∃y)(∃s)(x ∈ y ∧ φ(y, δ, s)). By Proposition 3.3.3 again, x ∈ Jδʹ for some δʹ < ...