Bounded arithmetic and truth definition

In [l], S. Buss introduced a system of Bounded Arithmetic S: which is closely related to A: in the polynomial hierachy. One of the most important open problems on Bounded Arithmetic is whether the hierachy of Bounded Arithmetic collapses, i.e., whether Si+?+’ = Sk for some i. This is relevant to the problem whether the polynomial hierchy collapses, though the exact logical relation between these two problems is not known at this moment. Very often, a stronger system is shown to be strictly stronger than a weaker system by proving that the stronger system proves the consistency of the weaker system. For the hierachy of Bounded Arithmetic this method does not work since Wilkie and Paris proved in [2] that Si + exp does not prove the consistency of Robinson’s arithmetic Q. where exp is the axiom Vx (2” exists). (S: + exp is equivalent to S: + exp.) This phenomenon can be explained by saying that the expressing power of unbounded quantifier is too strong to be handled by Bounded Arithmetic. Therefore the ordinary notion of consistency is totally inadequate for Bounded Arithmetic. We need some more delicate notions to distinguish systems of Bounded Arithmetic. For IZ = 1,2,3,. . . , we shall introduce Si,, which is obtained from S: by adding a function x #; y = 2’“’ # lYln where ]yln is defined by ]y],=y and IYI n+l = ]]yl,,]. We shall give a truth definition of Z+’ in S:,, for every 12 and prove that a Godel sentence of S: is provable in S$’ though it is not probable in Si. From this follows the existence of a A:-formula A(a) in S: which is not probable in Sg but provable in S$‘. Since S: can be considered as the limit of S:,, when n goes to ~0, this somewhat shows the delicate nature of the problem whether Si+?+’ is equivalent to Si. In [l], S. Buss also introduced second-order systems of Bounded Arithmetic fii and fii which are related to PSPACE and EXPTIME respectively. A formula in the language of S: is said to be sharply bounded if every quantifier in it is sharply bounded. We shall also define a truth definition of the sharply bounded formulas in @. By this truth definition, we can prove some weak consistency of 3: in @ where $’ is obtained from S: by adding the function /3(i, a) and by restricting all