Original version
Revolutions and Revelations in Computability -18th Conference on Computability in Europe, CiE 2022 Swansea, UK, July 11–15, 2022 Proceedings. 2022, 208-219, DOI: https://doi.org/10.1007/978-3-031-08740-0_18
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.