Bounded arithmetic in free logic (Q2904624)

The paper is devoted to the open problem whether Buss's hierarchy of theories of bounded arithmetic collapses or not. Buss's theories are reformulated using free logic. It is conjectured that such theories are easier to handle. To show this, it is proved that Buss's theories prove consistencies of induction-free fragments of introduced theories whose formulae have bounded complexity. It is also shown that theories in Buss's hierarchy can be interpreted in the introduced theories using a simple translation. Finally, finitistic Gödel sentences in the developed systems are investigated. The aim here is the hope of proving that a theory in a lower level of Buss's hierarchy cannot prove consistency of induction-free fragments of introduced theories whose formulae have higher complexity.