Girard’s maxim, that Peano Arithmetic is a theory of one inductive definition, is re-examined in the light of a weak theory EA formalising basic principles of Nelson’s predicative Arithmetic.

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)

In this article we study the systems KF and VF of truth over set theory as well as related systems and compare them with the corresponding systems over arithmetic.

The context for this paper is Feferman's theory of explicit mathematics, T 0 . We address a problem that was posed in [6]. Let MID be the principle stating that any monotone operation on classifications has a least fixed point. The main objective of this paper is to show that T 0 + MID, when based on classical logic, also proves the existence of non-monotone inductive definitions that arise from arbitrary extensional operations on classifications. From the latter we deduce (...) that MID, when adjoined to classical T 0 , leads to a much stronger theory than T 0. (shrink)

Extending Gödel's Dialectica interpretation, we provide a functional interpretation of classical theories of positive arithmetic inductive definitions, reducing them to theories of finite-type functionals defined using transfinite recursion on well-founded trees.

We study a new proof principle in the context of constructive Zermelo-Fraenkel set theory based on what we will call “non-deterministic inductive definitions”. We give applications to formal topology as well as a predicative justification of this principle.

In intuitionistic generalized predicative systems as constructive set theory, or constructive type theory, two categories have been proposed to play the role of the category of locales: the category FSp of formal spaces, and its full subcategory FSpi of inductively generated formal spaces. Considered in impredicative systems as the intuitionistic set theory IZF, FSp and FSpi are both equivalent to the category of locales. However, in the mentioned predicative systems, FSp fails to be closed under basic constructions such as that (...) of finite products, while FSpi is complete , but does not contain some naturally occurring classes of formal spaces. Moreover, completeness of FSpi only holds under the assumption of strong principles for the existence of inductively defined sets.In this paper, working in the context of constructive set theory, the category ImLoc of imaginary locales is introduced. ImLoc is complete already over weak formulations of constructive set theory, contains FSp and FSpi as full subcategories, and, as FSp and FSpi, is equivalent to the category of locales in IZF.Applications of the concept of imaginary locale to a proof of a point-free version of Tychonoff embedding theorem, and to set-representation theorems, are then presented. (shrink)

A new research project has, quite recently, been launched to clarify how different, from systems in second order number theory extending ACA0, those in second order set theory extending NBG are. In this article, we establish the equivalence between Δ01-LFP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta^1_0\mbox{\bf-LFP}}$$\end{document} and Δ01-FP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta^1_0\mbox{\bf-FP}}$$\end{document}, which assert the existence of a least and of a fixed point, respectively, for positive elementary operators. Our proof also shows (...) the equivalence between ID1 and ID^1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\widehat{\it ID}_1}$$\end{document}, both of which are defined in the standard way but with the starting theory PA replaced by ZFC. (shrink)

We locate winning strategies for various ${\mathrm{\Sigma }}_{3}^{0}$ -games in the L-hierarchy in order to prove the following: Theorem 1. KP+Σ₂-Comprehension $\vdash \exists \alpha L_{\alpha}\ models"\Sigma _{2}-{\bf KP}+\Sigma _{3}^{0}-\text{Determinacy}."$ Alternatively: ${\mathrm{\Pi }}_{3}^{1}\text{\hspace{0.17em}}-{\mathrm{C}\mathrm{A}}_{0}\phantom{\rule{0ex}{0ex}}$ "there is a β-model of ${\mathrm{\Delta }}_{3}^{1}-{\mathrm{C}\mathrm{A}}_{0}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\text{\hspace{0.17 em}}{\mathrm{\Sigma }}_{3}^{0}$ -Determinacy." The implication is not reversible. (The antecedent here may be replaced with ${\mathrm{\Pi }}_{3}^{1}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left({\mathrm{\Pi }}_{3}^{1}\right)-{\mathrm{C}\mathrm{A}}_{0}:\text{\hspace{0.17em}}{\mathrm{\Pi }}_{3}^{1}$ instances of Comprehension with only ${\mathrm{\Pi }}_{3}^{1}$ -lightface definable parameters—or even weaker theories.) Theorem 2. KP +Δ₂-Comprehension +Σ₂-Replacement + ${\mathrm{\Sigma }}_{3}^{0}\phantom{\rule{0ex}{0ex}}$ -Determinacy. (Here AQI (...) is the assertion that every arithmetical quasi-inductive definition converges.) Alternatively: $\Delta _{3}^{1}{\rm CA}_{0}+{\rm AQI}\nvdash \Sigma _{3}^{0}$ -Determinacy. Hence the theories: ${\mathrm{\Pi }}_{3}^{1}-{\mathrm{C}\mathrm{A}}_{0},\text{\hspace{0.17em}}{\mathrm{\Delta }}_{3}^{1}-{\mathrm{C}\mathrm{A}}_{0}+\text{\hspace{0.17em}}{\mathrm{\Sigma }}_{3}^{0}-\mathrm{D}\mathrm{e}\mathrm{t}\phantom{\rule{0ex}{0ex}}$ -Det, ${\mathrm{\Delta }}_{3}^{1}-{\mathrm{C}\mathrm{A}}_{0}+\mathrm{A}\mathrm{Q}\mathrm{I}$ , and ${\mathrm{\Delta }}_{3}^{1}-{\mathrm{C}\mathrm{A}}_{0}\phantom{\rule{0ex}{0ex}}$ are in strictly descending order of strength. (shrink)

In “Truth – A Traditional Debate Reviewed”, Crispin Wright proposed an inductive definition of “coherence truth” for arithmetic relative to an arithmetic base theory B. Wright’s definition is in fact a notational variant of the usual Tarskian inductive definition, except for the basis clause for atomic sentences. This paper provides a model-theoretic characterization of the resulting sets of sentences "cohering" with a given base theory B. These sets are denoted WB. Roughly, if B satisfies a (...) certain minimal condition, then WB is the Th, where M is the canonical model of the set At of atomic sentences provable in B. The paper also shows that the disquotational T-scheme is provable from Wright’s inductive definition just in case the base theory B is sound and complete for arithmetic atomic sentences. (shrink)

A fragment with the same provably recursive functions as n iterated inductive definitions is obtained by restricting second order arithmetic in the following way. The underlying language allows only up to n + 1 nested second order quantifications and those are in such a way, that no second order variable occurs free in the scope of another second order quantifier. The amount of induction on arithmetical formulae only affects the arithmetical consequences of these theories, whereas adding induction for arbitrary (...) formulae increases the strength by one inductive definition. (shrink)

In this paper, we prove the wellfoundedness of recursive notation systems for reflecting ordinals up to Π₃-reflection by relevant inductive definitions.

By forcing, we give a direct interpretation of into Avigad's. To the best of the author's knowledge, this is one of the simplest applications of forcing to “real problems”.

We interpret intuitionistic theories of (iterated) strictly positive inductive definitions (s.p.-ID i′ s) into Martin-Löf's type theory. The main purpose being to obtain lower bounds of the proof-theoretic strength of type theories furnished with means for transfinite induction (W-type, Aczel's set of iterative sets or recursion on (type) universes). Thes.p.-ID i′ s are essentially the wellknownID i -theories, studied in ordinal analysis of fragments of second order arithmetic, but the set variable in the operator form is restricted to occur (...) only strictly positively. The modelling is done by constructivizing continuity notions for set operators at higher number classes and proving that strictly positive set operators are continuous in this sense. The existence of least fixed points, or more accurately, least sets closed under the operator, then easily follows. (shrink)

In this paper first order theories for nonmonotone inductive definitions are introduced, and a proof-theoretic analysis for such theories based on combined operator forms a la Richter with recursively inaccessible and Mahlo closure ordinals is given.

This paper introduces theories for arithmetical quasi-inductive definitions (Burgess, 1986) as it has been done for first-order monotone and nonmonotone inductive ones. After displaying the basic axiomatic framework, we provide some initial result in the proof theoretic bounds line of research (the upper one being given in terms of a theory of sets extending Kripke–Platek set theory).

Exploiting the fact that -definable non-monotone inductive definitions have the same closure ordinal as arbitrary arithmetically definable monotone inductive definitions, we show that the proof theoretic ordinal of an axiomatization of -definable non-monotone inductive definitions coincides with the proof theoretic ordinal of the theory of arithmetically definable monotone inductive definitions.

We prove here that the intuitionistic theory $T_{0}\upharpoonright + UMID_{N}$ , or even $EEJ\upharpoonright + UMID_{N}$ , of Explicit Mathematics has the strength of $\prod_{2}^{1} - CA_{0}$ . In Section I we give a double-negation translation for the classical second-order $\mu-calculus$ , which was shown in [ $M\ddot{o}02$ ] to have the strength of $\prod_{2}^{1}-CA_{0}$ . In Section 2 we interpret the intuitionistic $\mu-calculus$ in the theory $EETJ\upharpoonright + UMID_{N}$ . The question about the strength of monotone inductive definitions (...) in $T_{0}$ was asked by S. Feferman in 1982, and - assuming classical logic - was addressed by M. Rathjen. (shrink)

We present statements equivalent to some fragments of the principle of non-deterministic inductive definitions (NID) by van den Berg (2013), working in a weak subsystem of constructive set theory CZF. We show that several statements in constructive topology which were initially proved using NID are equivalent to the elementary and finitary NIDs. We also show that the finitary NID is equivalent to its binary fragment and that the elementary NID is equivalent to a variant of NID based on the (...) notion of biclosed subset. Our result suggests that proving these statements in constructive topology requires genuine extensions of CZF with the elementary or finitary NID. (shrink)

1. Pohlers and The Problem. I first met Wolfram Pohlers at a workshop on proof theory organized by Walter Felscher that was held in Tübingen in early April, 1973. Among others at that workshop relevant to the work surveyed here were Kurt Schütte, Wolfram’s teacher in Munich, and Wolfram’s fellow student Wilfried Buchholz. This is not meant to slight in the least the many other fine logicians who participated there.2 In Tübingen I gave a couple of survey lectures on results (...) and problems in proof theory that had been occupying much of my attention during the previous decade. The following was the central problem that I emphasized there: The need for an ordinally informative, conceptually clear, proof-theoretic reduction of classical theories of iterated arithmetical inductive definitions to corresponding constructive systems. As will be explained below, meeting that need would be significant for the then ongoing efforts at establishing the constructive foundation for and proof-theoretic ordinal analysis of certain impredicative subsystems of classical analysis. I also spoke in Tübingen about. (shrink)

Gupta-Belnap-style circular definitions use all real numbers as possible starting points of revision sequences. In that sense they are boldface definitions. We discuss lightface versions of circular definitions and boldface versions of inductive definitions.

Definition is an important scientific and philosophical method. In all kinds of scientific and philosophical inquiries definition is provided to make clear the characteristics of the things under investigation. Definition in this sense, sometimes called real definition, should state the essence of the thing defined, according to Aristotle. In another (currently popular) sense, sometimes called nominal definition, definition explicates the meaning of a term already in use in an ordinary language or the scientific discourse (...) or specifies the meaning of a new term introduced in an ordinary language of the scientific discourse. Definition combines the purposes of both real and nominal definition and is promoted by the Nyaya philosophers of India. Another important method of science and philosophy is induction. In a narrow sense induction is a method of generalization to all cases from the observation of particular cases. In a broad sense induction is a method for reasoning from some observed fact to a different fact not involved in the former. We understand induction in the broad sense though more often we shall actually be concerned with induction in the narrow sense. How can our limited experience of nature provide the rational basis for making knowlege claims about unobserved phenomena? (shrink)

Gupta-Belnap-style circular definitions use all real numbers as possible starting points of revision sequences. In that sense they are boldface definitions. We discuss lightface versions of circular definitions and boldface versions of inductive definitions.

In this paper, I argue that there are cases of explanatory induction in mathematics. To do so, I first introduce the notion of explanatory definition in the context of mathematical explanation. A large part of the paper is dedicated to introducing and analyzing this notion of explanatory definition and the role it plays in mathematics. After doing so, I discuss a particular inductive definition in advanced mathematics—CW\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${ CW}$$\end{document}-complexes—and (...) argue that it is explanatory. With this, we see that there are cases of explanatory induction. (shrink)

This paper discusses Frege's account of definition by induction in Grundgesetze and the two key theorems Frege proves using it.