Weak Sequential Theories of Finite Full Binary Trees

Abstract

We study a first-order theory of finite full binary trees with an axiom schema of open induction. We show that this theory is sequential by constructing a direct interpretation of Adjunctive Set Theory in a very weak finitely axiomatized subtheory. We show that weakening the latter theory by removal of an axiom which states that the subtree relation is transitive gives a theory that directly interprets Vaught’s weak set theory, a non-finitely axiomatizable fragment of Adjunctive Set Theory.