Finiteness in not a ∑0-Property

We construct a sequence of transitive finite setsAn,n∈ω, such that, puttingA = Un∈ωAn,A is transitive of rank ω (without urelements) and, for every sentence Φ in the language of set theory,A ⊧φ if and only ifAn ⊧φ, for all but finitely manyn’s. This implies the claim of the title.