Forcing in finite structures

Abstract

We present a simple and completely model-theoretical proof of a strengthening of a theorem of Ajtai: The independence of the pigeonhole principle from IΔ0(R). With regard to strength, the theorem proved here corresponds to the complexity/proof-theoretical results of [10] and [14], but a different combinatorics is used. Techniques inspired by Razborov [11] replace those derived from Håstad [8]. This leads to a much shorter and very direct construction.