Cardinal Arithmetic in Weak Theories

In this paper we develop the theory of cardinals in the theory COPY. This is the theory of two total, jointly injective binary predicates in a second order version, where we may quantify over binary relations. The only second order axioms of the theory are the axiom asserting the existence of an empty relation and the adjunction axiom, which says that we may enrich any relation R with a pair x, y. The theory COPY is strictly weaker than the theory AS, adjunctive set theory. The relevant notion of weaker here is direct interpretability. We will explain and motivate this notion in the paper. A consequence is that our development of cardinals is inherited by stronger theories like AS. We will show that the cardinals satisfy (at least) Robinson’s Arithmetic Q. A curious aspect of our approach is that we develop cardinal multiplication using neither recursion nor pairing, thus diverging both from Frege’s paradigm and from the tradition in set theory. Our development directly uses the universal property characterizing the product that is familiar from category theory. The broader context of this paper is the study of a double degree structure: the degrees of (relative) interpretability and the finer degrees of direct interpretability. Most of the theories studied are in one of two degrees of interpretability: the bottom degree of predicate logic or the degree of Q. The theories will differ significantly if we compare them using direct interpretability.