On sets of relations definable by addition

Abstract

For every κ ∈ ω, there is an infinite set A_κ ⊆ ω and a d(κ) ∈ ω such that for all Q₀, Q₁, ⊆ A_κ where ∣Q₀∣ = ∣ Q₁∣, or d(κ) < ∣Q₀∣, Q₁∣ < ℵ₀, the structures ‹ω, +, Q₀› and ‹ω, +, Q₁› are indistinguishable by first-order sentences of quantifier depth κ whose atomic formulas are of the form u = v, u + v = w, and Q(u), where u, v, and w are variables.