In , S. Buss introduced the systems of Bounded Arithmetic for (i = 0,1,2,…) which has a close relationship to classes in polynomial hierarchy.
In , we defined a very special kind of proof-predicate Prfi for which gives detailed information on bounds of free variables used in the proof. There we also introduced infinitely many Gödel sentences for Prfi (k = 0, 1, 2, …) and showed that the properties of Prfi and are closely related to the P ≠ NP problem. Then we presented many conjectures on Prfi and which imply P ≠ NP.
Now in , Feferman emphasized that the arithmetization of metamathematics must be carried out intensionally. Bounded Arithmetic is a very interesting case in this sense.
In this paper, we also introduce the usual proof-predicate PRFi for and infinitely many Gödel sentences for PRFi(k= 0, 1, 2, …). Then we show that (Prfi, )and (PRFi, ) form a good contrast, this contrast is also closely related to the P ≠ NP problem, and present more conjectures which imply P ≠ NP.
As in  we define to be the following extension of Buss' original .
(1) We add finitely many function symbols which express polynomial time computable functions to Buss' original language of .
(2) All basic axioms on function symbols and ≤ can be expressed by initial sequents without logical symbols.