Let M be a transitive model of ZFC, and let φ be some arithmetical formula, then
ZFC⊢[ω⊨φ] implies M⊨[ω⊨φ] implies ω⊨φ
where the final implication holds since the ω of M is the real ω. More formally, we define:
x∈Ord iff [∀(y∈x), ∀(z∈y), z∈x & ∀(w∈z),w∈y]
x∈Lim iff [x∈Ord & x≠∅ & ∀(y∈x), ∃(z∈x), y∈z]
x∈ω iff [x∈Ord & x∉Lim & ∀(y∈x), y∉Lim]
We easily show that the condition x∈ω, as defined above, is a Δ0 property, which is absolute for transitive sets. That is, if M is a transitive set and ψ is a Δ0 formula in the language of sets, then ψ⇔[M⊨ψ]. This means that x∈ω⇔M⊨[x∈ω], for all x∈M. We e…