The bounded arithmetic theory S 2 is finitely axiomatized if and only if the polynomial hierarchy provably collapses. If T 2 equals S then T 2 is equal to S 2 and proves that the polynomial time hierarchy collapses to # , and, in fact, to the Boolean hierarchy over # and to # i+1 /poly .

This paper presents a new proof of the characterization of the provably recursive functions of the fragments I# n of Peano arithmetic. The proof method also characterizes the # k -definable functions of I# n and of theories axiomatized by transfinite induction on ordinals. The proofs are completely proof-theoretic and use the method of witness functions and witness oracles.

This paper investigates the provably total functions of fragments of first- and second-order Bounded Arithmetic. The (strongly) 3 are precisely the (strong) ] functions. The # and U are the EXPTIME [wit, poly] functions and the # definable functions of V 2 are the EXPTIME -functions. We give witnessing theorems for these theories and prove conservation results 3 over S 3 and for U 2 over V 2 .