A non-sequential arithmetical theory with pairing

Abstract

Albert Visser has shown that Robinson's $ \mathsf{Q} $ and Gregorczyk's $ \mathsf{TC} $ are not sequential by showing that these theories are not even poly-pair theories, which, in a strong sense, means these theories lack pairing. In this paper, we use Ehrenfeucht-Fra\"iss\'e games to show that the theory $ \mathsf{Q} + \Theta $ we obtain by extending Robinson's $ \mathsf{Q} $ with an axiom $ \Theta $ which says that the map $ \pi (x, y ) = (x+y)^2 + x $ is a pairing function is not sequential; in fact, we show that this theory is not even a Vaught theory. As a corollary, we get that the tree theory $ \mathsf{T} $ of [Kristiansen & Murwanashyaka, 2020] is also not a Vaught theory.