We characterize the proof-theoretic strength of systems of explicit mathematics with a general principle asserting the existence of least fixed points for monotone inductive definitions, in terms of certain systems of analysis and set theory. In the case of analysis, these are systems which contain the Σ12-axiom of choice and Π12-comprehension for formulas without set parameters. In the case of set theory, these are systems containing the Kripke-Platek axioms for a recursively inaccessible universe together with the existence of a stable (...) ordinal. In all cases, the exact strength depends on what forms of induction are admitted in the respective systems. (shrink)

KPM is a subsystem of set theory designed to formalize a recursively Mahlo universe of sets. In this paper we show that a certain ordinal notation system is sufficient to measure the proof-theoretic strength ofKPM. This involves a detour through an infinitary calculus RS(M), for which we prove several cutelimination theorems. Full cut-elimination is available for derivations of $\Sigma (L_{\omega _1^c } )$ sentences, whereω 1 c denotes the least nonrecursive ordinal. This paper is self-contained, at least from a technical (...) point of view. (shrink)

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.

The Suslin operator is a type-2 functional testing for the well-foundedness of binary relations on the natural numbers. In the context of applicative theories, its proof-theoretic strength has been analyzed in Jäger and Strahm [18]. This article provides a more direct approach to the computation of the upper bounds in question. Several theories featuring the Suslin operator are embedded into ordinal theories tailored for dealing with non-monotone inductive definitions that enable a smooth definition of the application relation.

In the first part we show why ordinals and ordinal notations are naturally connected with proof theoretical research. We introduce the program of ordinal analysis. The second part gives examples of applications of ordinal analysis.

We give an overview of recent results in ordinal analysis. Therefore, we discuss the different frameworks used in mathematical proof-theory, namely "subsystem of analysis" including "reverse mathematics", "Kripke-Platek set theory", "explicit mathematics", "theories of inductive definitions", "constructive set theory", and "Martin-Löf's type theory".

We study the extension of Feferman’s operational set theory provided by adding operational versions of unbounded existential quantification and power set and determine its proof-theoretic strength in terms of a suitable theory of sets and classes.

We study and some of its most important extensions primarily from a proof-theoretic perspective, determine their consistency strengths by exhibiting equivalent systems in the realm of traditional set theory and introduce a new and interesting extension of which is conservative over.

We give a broad discussion of reflection principles in explicit mathematics, thereby addressing various kinds of universe existence principles. The proof-theoretic strength of the relevant systems of explicit mathematics is couched in terms of suitable extensions of Kripke–Platek set theory.

In this article we introduce and study the notion of operational closure: a transitive set d is called operationally closed iff it contains all constants of OST and any operation f∈d applied to an element a∈d yields an element fa∈d, provided that f applied to a has a value at all. We will show that there is a direct relationship between operational closure and stability in the sense that operationally closed sets behave like Σ1 substructures of the universe. This leads (...) to our final result that OST plus the axiom , claiming that any set is element of an operationally closed set, is proof-theoretically equivalent to the system KP+ of Kripke–Platek set theory with infinity and Σ1 separation. We also characterize the system OST plus the existence of one operationally closed set in terms of Kripke–Platek set theory with infinity and a parameter-free version of Σ1 separation. (shrink)

In this paper we introduce a recursive notation system $O(\mu)$ of ordinals. An element of the notation system is called an ordinal diagram following G. Takeuti [25]. The system is designed for proof theoretic study of theories of recursively Mahlo universes. We show that for each $\alpha<\Omega$ in $O(\mu)$ KPM proves that the initial segment of $O(\mu)$ determined by $\alpha$ is a well ordering. Proof theoretic study for such theories will be reported in [9].

In this article we show how to use the result in Jäger and Probst [7] to adapt the technique of pseudo-hierarchies and its use in Avigad [1] to subsystems of set theory without foundation. We prove that the theory KPi0 of admissible sets without foundation, extended by the principle (Σ-FP), asserting the existence of fixed points of monotone Σ operators, has the same proof-theoretic ordinal as KPi0 extended by the principle (Σ-TR), that allows to iterate Σ operations along ordinals. By (...) Jäger and Probst [6] we conclude that the metapredicative Mahlo ordinal φ ω00 is also the ordinal of KPi0+(Σ-FP). Hence the relationship between fixed points and iteration persists in the framework of set theory without foundation. (shrink)

In this paper we introduce theories of universes in analysis. We discuss a non-uniform, a uniform and a minimal variant. An analysis of the proof-theoretic bounds of these systems is given, using only methods of predicative proof-theory. It turns out that all introduced theories are of proof-theoretic strength between Γ0 and ϕ1ɛ00.

We propose two admissible closures $${\mathbb{A}({\sf PTCA})}$$ and $${\mathbb{A}({\sf PHCA})}$$ of Ferreira’s system PTCA of polynomial time computable arithmetic and of full bounded arithmetic (or polynomial hierarchy computable arithmetic) PHCA. The main results obtained are: (i) $${\mathbb{A}({\sf PTCA})}$$ is conservative over PTCA with respect to $${\forall\exists\Sigma^b_1}$$ sentences, and (ii) $${\mathbb{A}({\sf PHCA})}$$ is conservative over full bounded arithmetic PHCA for $${\forall\exists\Sigma^b_{\infty}}$$ sentences. This yields that (i) the $${\Sigma^b_1}$$ definable functions of $${\mathbb{A}({\sf PTCA})}$$ are the polytime functions, and (ii) the $${\Sigma^b_{\infty}}$$ definable (...) functions of $${\mathbb{A}({\sf PHCA})}$$ are the functions in the polynomial time hierarchy. (shrink)