About the proof-theoretic ordinals of weak fixed point theories

Abstract

Abstract. This paper presents several proof-theoretic results concerning weak fixed point theories over second order number theory with arithmetic comprehension and full or restricted induction on the natural numbers. It is also shown that there are natural second order theories which are proof-theoretically equivalent but have different proof-theoretic ordinals.Fixed points and fixed point theories play an important role in many branches of mathematical logic and theoretical computer science. The spectrum ranges from the fixed point theorem in recursion theory to fixed point arguments in categorical logic and denotational semantics of high-level programming languages.In proof theory, special emphasis has been put on formal systems for fixed points of (iterated) inductive definitions and their relationship to subsystems of analysis, set theory and constructive mathematics (cf. e.g. Buchholz, Feferman, Pohlers and Sieg [2], Feferman [3], and Jager [7]). However, interpreted in the proper sense, also features of modern type theories can be studied in terms of fixed points, and many concepts in nonmonotonic reasoning (circumscription, completion of theories, etc.) are related to fixed point theories. To a certain extent even parts of logic programming are built upon fixed point constructions (cf. e.g. Lloyd [9] and Jager and Stark [8]).The general purpose of this paper is to study several proof-theoretic aspects of the fixed point theories FP-ACA 0 and FP-ACA. They are formulated in the language of second order arithmetic, contain the axioms of primitive recursive arithmetic PRA, and comprise comprehension for arithmetic formulas. In addition, there has to be a fixed point for every positive arithmetic definition clause, and this fixed point can be proved to define a set. Both theories differ in the principles of complete induction which are available: In FP-ACA 0 complete induction on the natural numbers is restricted to sets, whereas FP-ACA contains complete induction for arbitrary formulas.We will show that the proof-theoretic ordinals of FP-ACA 0 and FP-ACA are £ 0 and <pls 0 , respectively. This is remarkable since we will also see that FP-ACA 0 is proof-theoretically equivalent to the system (E\-AC) of second order arithmetic,

Non‐elementary speed‐ups in logic calculi

“…Next by an asymmetric interpretation as in [5], assign a pair of natural numbers (n, m) (n ≤ m) to occurrences of sequents in P . When a pair (n, m) is assigned to a sequent Γ, then replace any negative occurrence of I ϕ (positive occurrence of ¬I ϕ ) in Γ by ¬I n ϕ , and any positive occurrence of I ϕ by I m ϕ .…”

“…Let Tr(x) denote a formula which says that x is a true sentence in the language L 1 of EA, in the standard model: (5) T "the sentence EA is true with respect to Tr" and T "Tr enjoys Tarski's conditions".…”