Bounded Arithmetic in Free Logic

One of the central open questions in bounded arithmetic is whether Buss’ hierarchy of theories of bounded arithmetic collapses or not. In this paper, we reformulate Buss’ theories using free logic and conjecture that such theories are easier to handle. To show this, we first prove that Buss’ theories prove consistencies of induction-free fragments of our theories whose formulae have bounded complexity. Next, we prove that although our theories are based on an apparently weaker logic, we can interpret theories in Buss’ hierarchy by our theories using a simple translation. Finally, we investigate finitistic Gödel sentences in our systems in the hope of proving that a theory in a lower level of Buss’ hierarchy cannot prove consistency of induction-free fragments of our theories whose formulae have higher complexity.