Finiteness in not a ∑₀-Property

We construct a sequence of transitive finite sets A _n, n∈ω, such that, putting A = U _n∈ω A _n, A is transitive of rank ω (without urelements) and, for every sentence Φ in the language of set theory, A {models}φ if and only if A _n {models}φ, for all but finitely many n's. This implies the claim of the title.