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, wereformulate Buss' theories using free logic and conjecture that such theoriesare easier to handle. To show this, we first prove that Buss' theories proveconsistencies of induction-free fragments of our theories whose formulae havebounded complexity. Next, we prove that although our theories are based on anapparently weaker logic, we can interpret theories in Buss' hierarchy by ourtheories using a simple translation. Finally, we investigate finitistic G\"odelsentences in our systems in the hope of proving that a theory in a lower levelof Buss' hierarchy cannot prove consistency of induction-free fragments of ourtheories whose formulae have higher complexity