On the ∀∃-Theories of Free Projective Planes

Abstract

Studying the elementary properties of free projective planes of finite rank, we prove that for m > n, an arbitrary ∀∃∀-formula Φ(ȳ) and a tuple ū of elements of the free projective plane $$\mathfrak{F}_{n}$$ if Φ(ū) holds on the plane $$\mathfrak{F}_{m}$$ then Φ(ū) holds on the plane $$\mathfrak{F}_{n}$$ too. This implies the coincidence of the ∀∃-theories of free projective planes of different finite ranks.