The first example of a simultaneous inductive-recursive definition in intuitionistic type theory is Martin-Löf's universe à la Tarski. A set U0 of codes for small sets is generated inductively at the same time as a function T0 , which maps a code to the corresponding small set, is defined by recursion on the way the elements of U0 are generated. In this paper we argue that there is an underlying general notion of simultaneous inductiverecursive definition which is implicit in Martin-Löf's intuitionistic type theory. We extend previously given schematic formulations of inductive definitions in type theory to encompass a general notion of simultaneous induction-recursion. This enables us to give a unified treatment of several interesting constructions including various universe constructions by Palmgren, Griffor, Rathjen, and Setzer and a constructive version of Aczel's Frege structures. Consistency of a restricted version of the extension is shown by constructing a realisability model ...

Contents 1 Introduction 1-1 2 Some Axiom Systems 2-1 2.1 Classical Set Theory . . . . . . . . . . . . . . . . . . . . . . . 2-1 2.2 Intuitionistic Set Theory . . . . . . . . . . . . . . . . . . . . . 2-2 2.3 Constructive Set Theory . . . . . . . . . . . . . . . . . . . . . 2-2 2.3.1 CZF 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-2 2.3.2 Constructive Zermelo Fraenkel, CZF . . . . . . . . . . 2-3 2.3.3 Basic Constructive Set Theory, BCST . . . . . . . . . 2-3 3 Elementary Mathematics in Constructive Set Theory 3-1 3.1 Class Notation . . . . . . . . . . . . . . . . . . . . . . . . . . 3-1 3.2 Pairing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-2 3.2.1 Ordered Pairs . . . . . . . . . . . . . . . . . . . . . . . 3-2 3.2.2 Cartesian Products of Classes . . . . . . . . . . . . . . 3-2 3.2.3 Relations and Functions between Classes . . . . . . . . 3-3 3.3 The class of Natural

Induction-recursion is a schema which formalizes the principles for introducing new sets in Martin-Löf’s type theory. It states that we may inductively define a set while simultaneously defining a function from this set into an arbitrary type by structural recursion. This extends the notion of an inductively defined set substantially and allows us to introduce universes and higher order universes (but not a Mahlo universe). In this article we give a finite axiomatization of inductive-recursive definitions. We prove consistency by constructing a set-theoretic model which makes use of one Mahlo cardinal. 1

The notion of a universe of types was introduced into constructive type theory by Martin-Löf (1975). According to the propositions-as-types principle inherent in

We set out to study the consequences of the assumption of types of wellfounded trees in dependent type theories. We do so by investigating the categorical notion of wellfounded tree introduced in [16]. Our main result shows that wellfounded trees allow us to define initial algebras for a wide class of endofunctors on locally cartesian closed categories.

We present well-ordering proofs for Martin-Lof's type theory with W-type and one universe. These proofs, together with an embedding of the type theory in a set theoretical system as carried out in [Set93] show that the proof theoretical strength of the type theory is precisely ## 1# I+# , which is slightly more than the strength of Feferman's theory T 0 , classical set theory KPI and the subsystem of analysis (# 1 2 -CA)+(BI). The strength of intensional and extensional version, of the version a la Tarski and a la Russell are shown to be the same. 0 Introduction 0.1 Proof theory and Type Theory Proof theory and type theory have been two answers of mathematical logic to the crisis of the foundations of mathematics at the beginning of the century. Proof theory was originally established by Hilbert in order to prove the consistency of theories by using finitary methods. When Godel showed that Hilbert's program cannot be carried out as originally intended, the focus of proof theory ch...

this paper we show how to combine the unrestricted countable choice, induction on infinite well-founded trees and restricted classical logic in a constructively given model. For readers faniliar with intuitionistic systems [14], we may succinctly describe the theory we interpret as follows. Expand the extensional version of HA

We define a type theory MLM, which has proof theoretical strength slightly greater then Rathjens theory KPM. This is achieved by replacing the universe in Martin-Lof's Type Theory by a new universe V, which has the property that for every function f , mapping families of sets in V to families of sets in V, there exists a universe closed under f . We show that the proof theoretical strength of MLM is /\Omega 1\Omega M+! . Therefore we reach a strength slightly greater than jKPMj and V can be considered as a Mahlo-universe. Together with [Se96a] it follows jMLMj = /\Omega 1(\Omega M+! ). 1 Introduction An ordinal M is recursively Mahlo iff M is admissible and every M-recursive closed unbounded subset of M contains an admissible ordinal. Equivalently, this is the case iff M is admissible and for all \Delta 0 formulas OE(x; y; ~z), and all ~z 2 LM such that 8x 2 LM :9y 2 LM :OE(x; y; ~z) there exists an admissible ordinal fi ! M such that 8x 2 L fi 9y 2 L fi :OE(x; y; ~z) holds. ...

Abstract. We introduce a new kind of models for constructive set theories based on categories of presheaves. These models are a counterpart of the presheaf models for intuitionistic set theories defined by Dana Scott in the ’80s. We also show how presheaf models fit into the framework of Algebraic Set Theory and sketch an application to an independence result. 1. Variable sets in foundations and practice Presheaves are of central importance both for the foundations and the practice of mathematics. The notion of a presheaf formalizes well the idea of a variable set, that is relevant in all the areas of mathematics concerned with the study of indexed families of objects [19]. One may then readily see how presheaves are of interest also in foundations: both Cohen’s forcing models for classical set theories and Kripke models for intuitionistic logic involve the idea of sets indexed by stages. Constructive aspects start to emerge when one considers the internal logic of categories of presheaves. This logic, which does not include classical principles such as the law of the excluded middle, provides a useful language to manipulate objects

In this paper we discuss extensions of Feferman's theory T_0 for explicit mathematics by the so-called limit and Mahlo axioms and present a novel approach to constructing natural recusion-theoretic models for (fairly strong) systems of explicit mathematics which is based on nonmonotone inductive definitions.