On sets of relations definable by addition
James Francis Lynch · Journal of Symbolic Logic · 1982
Abstract For every κ ∈ ω, there is an infinite set A κ ⊆ ω and a d (κ) ∈ ω such that for all Q 0 , Q 1 , ⊆ A κ where ∣ Q 0 ∣ = ∣ Q 1 ∣, or d (κ) < ∣ Q 0 ∣, Q 1 ∣ < ℵ 0 , the structures ‹ω, +, Q 0 › and ‹ω, +, Q 1 › 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.