On families of testing formulae for a pp formula

Abstract

Astier and Tressl have recently proven that a pp formula fails on a finite subspace of a space of orderings if and only if a certain family of formulae is verified (V. Astier, M. Tressl, Axiomatization of localglobal principles for pp formulas in spaces of orderings, Arch. Math. Logic 44, No. 1 (2005), 77-95). The proof given in their paper is nonconstructive and uses rather advanced techniques from model theory. In this note we slightly strengthen their result by constructing another family of formulae with the same property, whose elements are given explicitly. We also illustrate the developed theory with an example of the testing family for a pp formula that is known to be a counterexample to the pp conjecture.