From Sets and Types to Topology and Analysis

This introductory chapter sets the stage for the following chapters. It first discusses the issue of foundational systems for constructive mathematical practice, and expresses the desire for a closer interplay between foundations and practice of constructive mathematics as they seem to have parted from each other in more recent times. It observes that a closer connection is already witnessed by some of the contributions to this book. With the intent of fostering the discussion on foundations, it suggests some ideas on (weak) foundational systems for constructive mathematics. Finally, it recalls the basic steps in the fields of constructive set and type theory, formal topology, and constructive analysis, and gives a presentation of the contributions in the volume to each field.

This paper describes a fibrational categorical semantics for the modal necessity-only fragment of constructive modal type theory, both with and without dependent types. Constructive type theory does not usually discuss logical modalities, and modalities tend to be mostly studied within classical logic, not within type theory. But modalities should be very useful in type theory, as they are very useful in modelling theoretical computing systems. Providing constructive versions of modal logics and their associated Curry-Howard modal type theories is also a very productive program, e.g. helpful when dealing with computational effects, staged computation, and functional reactive types, for example. There seems to be renewed interest in the notion of constructive modal type theory (and in notions of linear type theory), in part because of the interest in homotopy type theory. The modal type theory presented here uses dependent types, in the style of Ritter's categorical models of the Calculus of Constructions. To build up to these, we first discuss the kinds of constructive modal type theory in the literature. Then we provide a non-dependent modal type theory, introduced in previous work, that we generalize to dependent types in the following section. Dependent type theories are usually but not always given categorical semantics in terms of fibrations. We provide semantics in terms of fibrations for both the non-dependent and the dependent type systems discussed and prove them sound and complete, thereby providing evidence that the type theory is meaningful. These fibrational models should be also applicable to the homotopy type theory setting.

Quotient topologies in constructive set theory and type theory

We propose an extension of Aczel's constructive set theory CZF by an axiom for inductive types and a choice principle, and show that this extension has the following properties: it is interpretable in Martin-Löf's type theory (hence acceptable from a constructive and generalized-predicative standpoint). In addition, it is strong enough to prove the Set Compactness theorem and the results in formal topology which make use of this theorem. Moreover, it is stable under the standard constructions from algebraic set theory, namely exact completion, realizability models, forcing as well as more general sheaf extensions. As a result, methods from our earlier work can be applied to show that this extension satisfies various derived rules, such as a derived compactness rule for Cantor space and a derived continuity rule for Baire space. Finally, we show that this extension is robust in the sense that it is also reflected by the model constructions from algebraic set theory just mentioned.

This is the first of two articles dedicated to the notion of constructive set. In them we attempt a comparison between two different notions of set which occur in the context of the foundations for constructive mathematics. We also put them under perspective by stressing analogies and differences with the notion of set as codified in the classical theory Zermelo–Fraenkel. In the current article we illustrate in some detail the notion of set as expressed in Martin–L¨of type theory and present the essential characters of this theory. In a second article we shall explore a distinct notion of set, as arising in the context of intuitionistic versions of Zermelo–Fraenkel set theory. The theory we shall analyse there is Aczel’s CZF (Constructive Zermelo– Fraenkel) and we shall supplement its exposition by a succinct account of Aczel’s interpretation of CZF in type theory. This will enable us to compare the two notions in a more precise sense.

Homotopy Type Theory (HoTT) is a putative new foundation for mathematics grounded in constructive intensional type theory, that offers an alternative to the foundations provided by ZFC set theory and category theory. This paper explains and motivates an account of how to define, justify and think about HoTT in a way that is self-contained, and argues that it can serve as an autonomous foundation for mathematics. We first consider various questions that a foundation for mathematics might be expected to answer, and find that the standard formulation of HoTT as presented in the ``HoTT Book'' does not answer many of them. More importantly, the way HoTT is developed in the HoTT Book suggests that it is not a candidate \emph{autonomous} foundation since it explicitly depends upon other fields of mathematics, in particular homotopy theory. We give an alternative presentation of HoTT that does not depend upon sophisticated ideas from other parts of mathematics, and in particular makes no reference to homotopy theory (but is compatible with the homotopy interpretation). Our elaboration of HoTT is based on a new interpretation of types as \emph{mathematical concepts}, which accords with the intensional nature of the type theory.

Topological inductive definitions

A Simple Nominal Type Theory

Metapredicativity is a new general term in proof theory which describes the analysis and study of formal systems whose proof-theoretic strength is beyond the Feferman-Schutte ordinal Γ0 but which are nevertheless amenable to purely predicative methods. Typical examples of formal systems which are apt for scaling the initial part of metapredicativity are the transfinitely iterated fixed point theories IDα whose detailed proof-theoretic analysis is given by Jager, Kahle, Setzer and Strahm in [18]. In this paper we assume familiarity with [18]. For natural extensions of Friedman’s ATR that can be measured against transfinitely iterated fixed point theories the reader is referred to Jager and Strahm [20]. In the mid seventies, Feferman [3, 4] introduced systems of explicit mathematics in order to provide an alternative foundation of constructive mathematics. More precisely, it was the origin of Feferman’s program to give a logical account of Bishop-style constructive mathematics. Right from the beginning, systems of explicit mathematics turned out to be of general interest for proof theory, mainly in connection with the proof-theoretic analysis of subsystems of first and second order arithmetic and set theory, cf. e.g. Jager [15] and Jager and Pohlers [19]. More recently, systems of explicit mathematics have been used to develop a general logical framework for functional programming and type theory, where it is possible to derive correctness and termination properties of functional programs. Important references in this connection are Feferman [6, 7, 9] and Jager [17]. Universes are a frequently studied concept in constructive mathematics at least since the work of Martin-Lof, cf. e.g. Martin-Lof [23] or Palmgren [27] for

Inaccessible set axioms may have little consistency strength

Constructive toposes with countable sums as models of constructive set theory

The authors present a proof development system called PowerEpsilon, based on a constructive type theory which can be used as a formal program development system for deriving a program from a specification. PowerEpsilon is a polymorphic language based on Martin-Lof's type theory and the calculus of constructions. In PowerEpsilon, the concept of limit of type universe hierarchies and a scheme for inductive define types are introduced. The system can be used as both a programming language with a very rich set of data structures and a metalanguage for formalizing constructive mathematics. The system has been implemented. A programming exercise is given to show how the system works. >

The aim of this book is to show how constructive type theory, in the form developed by Per Martin-Löf, helps us to understand the structure of natural language. The study of similar questions within simple (=classical) type theory was inaugurated by Frege and given a standard of technical detail by Montague. The point of shifting to constructive type theory is not in the constructivist criticism against classical reasoning, but in the positive observation that constructive type theory has new things to say. Some of the problems to which grammars based on classical type theory have not found satisfactory solutions will be shown to require precisely these new things. These problems appear in a simple form in what is known, since Geach, as donkey sentences. But they are repeated in other guises, for example, in the functioning of tenses, and in a larger scale, such as in the progressive structure of texts, and in the significance of context in communication.

We perform a proof-theoretical investigation of two modal predicate logics: global intuitionistic logic GI and global intuitionistic fuzzy logic GIF. These logics were introduced by Takeuti and Titani to formulate an intuitionistic set theory and an intuitionistic fuzzy set theory together with their metatheories. Here we define analytic Gentzen style calculi for GI and GIF. Among other things, these calculi allows one to prove Herbrand’s theorem for suitable fragments of GI and GIF.

In this article we continue the work started in [Ernesto Copello, Álvaro Tasistro, Nora Szasz, Ana Bove, and Maribel Fernández. Alpha-structural induction and recursion for the λ-calculus in constructive type theory. Electronic Notes in Theoretical Computer Science, 323:109–124, 2016], deriving in Constructive Type Theory new induction principles for the λ-calculus, using (the historical) first order syntax with only one sort of names for both bound and free variables, and with α-conversion based upon name swapping. The principles provide a flexible framework for mimicking pen-and-paper proofs within the rigorous formal setting of a proof assistant. We here report on one successful application, namely a complete proof of the Church-Rosser Theorem. The whole development has been machine-checked using the system Agda [Ulf Norell. Towards a Practical Programming Language Based on Dependent Type Theory. PhD thesis, Department of Computer Science and Engineering, Chalmers University of Technology, September 2007].