Jäger, G., Fixed points in Peano arithmetic with ordinals, Annals of Pure and Applied Logic 60 119-132. This paper deals with some proof-theoretic aspects of fixed point theories over Peano arithmetic with ordinals. It studies three such theories which differ in the principles which are available for induction on the natural numbers and ordinals. The main result states that there is a natural theory in this framework which is a conservative extension of Peano arithmeti.

Feferman, S. and G. Jäger, Systems of explicit mathematics with non-constructive μ-operator. Part I, Annals of Pure and Applied Logic 65 243-263. This paper is mainly concerned with the proof-theoretic analysis of systems of explicit mathematics with a non-constructive minimum operator. We start off from a basic theory BON of operators and numbers and add some principles of set and formula induction on the natural numbers as well as axioms for μ. The principal results then state: BON plus set induction (...) is proof-theoretically equivalent to Peano arithmetic PA; BON plus formula induction is proof-theoretically equivalent to the system <0 of second-order arithmetic. (shrink)

In this paper we study applicative theories of operations and numbers with the non-constructive minimum operator in the context of a total application operation. We determine the proof-theoretic strength of such theories by relating them to well-known systems like Peano Arithmetic PA and the system <0 of second order arithmetic. Essential use will be made of so-called fixed-point theories with ordinals, certain infinitary term models and Church-Rosser properties.

In this thesis, we study the least fixed point principle in a constructive setting. A constructive theory of functions and sets has been developed by Feferman. This theory deals both with sets and with functions over sets as independent notions. In the language of Feferman's theory, we are able to formulate the least fixed point principle for monotone inductive definitions as: every operation on classes to classes which satisfies the monotonicity condition has a least fixed point. This is called the (...) principle of monotone inductive definition. Furthermore, we may formulate this principle in a uniform way as: there is an operation which maps a monotone operation to its least fixed point. This is called the principle of uniform monotone inductive definition. Feferman raised the question of the strength of the principle of monotone inductive definition when adjoined to his theory . This question is our primary concern in this thesis. ;Our main results are the consistency of both the principle of monotone inductive definition and the principle of uniform monotone inductive definition adjoined to his theory, and the proof theoretical equivalence between the principle of monotone inductive definition with Feferman's system without the inductive generation axiom and the system of the Pi-one-one Comprehension Axiom. Determination of the proof-theoretical strength of the principle of monotone inductive definition adjoined to Feferman's theory still remains open. ;The consistency of both the principle of monotone inductive definition and the principle of uniform monotone inductive definition with Feferman's theory will be achieved by constructing their models in set theoretical sense. The proof theoretical equivalence between the principle of monotone inductive definition with Feferman's system without the inductive generation axiom and the system of the Pi-one-one Comprehension Axiom can be obtained by a careful examination of the model construction for the principle of monotone inductive definition with Feferman's system without the inductive generation axiom, which is parallel to the model construction for the principle of monotone inductive definition with his theory. (shrink)

We give a new elementary proof of the comparison theorem relating $\sum^1_{n + 1}-\mathrm{AC}\uparrow$ and $\Pi^1_n -\mathrm{CA}\uparrow$ ; the proof does not use Skolem theories. By the same method we prove: a) $\sum^1_{n + 1}-\mathrm{DC} \uparrow \equiv (\Pi^1_n -CA)_{ , for suitable classes of sentences; b) $\sum^1_{n+1}-DC \uparrow$ proves the consistency of (Π 1 n -CA) ω k, for finite k, and hence is stronger than $\sum^1_{n+1}-AC \uparrow$ . a) and b) answer a question of Feferman and Sieg.

The aim of this paper is the analysis of uniformity in Feferman's explicit mathematics. The proof-strength of those systems for constructive mathematics is determined by reductions to subsystems of second-order arithmetic: If uniformity is absent, the method of standard structures yields that the strength of the join axiom collapses. Systems with uniformity and join are treated via cut elimination and asymmetrical interpretations in standard structures.

Stephen George Simpson. with definition 1.2.3 and the discussion following it. For example, taking 90(n) to be the formula n §E Y, we have an instance of comprehension, VYEIXVn(n€X<—>n¢Y), asserting that for any given set Y there exists a ...

We study elementary second order extensions of the theoryID 1 of non-iterated inductive definitions and the theoryPA Ω of Peano arithmetic with ordinals. We determine the exact proof-theoretic strength of those extensions and their natural subsystems, and we relate them to subsystems of analysis with arithmetic comprehension plusΠ 1 1 comprehension and bar induction without set parameters.

This paper is mainly concerned with proof-theoretic analysis of some second-order systems of explicit mathematics with a non-constructive minimum operator. By introducing axioms for variable types we extend our first-order theory BON to the elementary explicit type theory EET and add several forms of induction as well as axioms for μ. The principal results then state: EET plus set induction is proof-theoretically equivalent to Peano arithmetic PA <0).