Peter Aczel

An Introduction to Inductive Definitions

Publisher Summary Inductive definitions of sets are often informally presented by giving some rules for generating elements of the set and then adding that an object is to be in the set only if it has been generated according to the rules. An equivalent formulation is to characterize the set as the smallest set closed under the rules. This chapter discusses monotone induction and its role in extensions of recursion theory. The chapter reviews some of the work on non-monotone induction and outlines the separate motivation that has led to its development. The chapter briefly considers inductive definitions in a more general context.

A Final Coalgebra Theorem

We prove that every set-based functor on the category of classes has a final coalgebra. This result strengthens the final coalgebra theorem announced in the book “Non-well-founded Sets”, by the first author.

The Type Theoretic Interpretation of Constructive Set Theory

By adding to Martin-LSfs intuitionistic theory of types a ‘type of sets’ we give a constructive interpretation of constructive set theory. constructive version of the classical conception of the cumulative hierarchy of sets. This interpretation is a

Frege Structures and the Notions of Proposition, Truth and Set

The notion of Frege structure is introduced and shown to give a coherent context for the rigorous development of Freges logical notion of set and an explanation of Russells paradox.

The Type Theoretic Interpretation of Constructive Set Theory: Choice Principles

In an earlier paper I gave an interpretation of a system CZF of constructive set theory within an extension of Martin-Lofs intuitionistic theory of types. In this paper some additional axioms, each a consequence of the axiom of choice, are shown to hold in the interpretation. The mathematical deductions are presented in an informal, but I hope rigorous style.

Infinite trees and completely iterative theories: a coalgebraic view

Infinite trees form a free completely iterative theory over any given signature--this fact, proved by Elgot, Bloom and Tindell, turns out to be a special case of a much more general categorical result exhibited in the present paper. We prove that whenever an endofunctor H of a category has final coalgebras for all functors H(-) + X, then those coalgebras, TX, form a monad. This monad is completely iterative, i.e., every guarded system of recursive equations has a unique solution. And it is a free completely iterative monad on H. The special case of polynomial endofunctors of the category Set is the above mentioned theory, or monad, of infinite trees.This procedure can be generalized to monoidal categories satisfying a mild side condition: if, for an object H, the endofunctor H ⊗ _ + I has a final coalgebra, T, then T is a monoid. This specializes to the above case for the monoidal category of all endofunctors.

The Type Theoretic Interpretation of Constructive Set Theory: Inductive Definitions

Publisher Summary Constructive set theory is a possible framework for the formalization of constructive mathematics. The chapter describes the formal system CZF+DC and its type theoretic interpretation. An inductive definition usually involves the characterization of a collection of objects as the smallest collection satisfying certain closure conditions. Such a characterization can be made explicit in one of at least two ways. The first way is to define the collection as the intersection of all collections that satisfies the closure conditions. Such an explicit definition is thoroughly impredicative in that the collection is defined using quantification over all collections. The second way is to build up the collection from below as the union of a hierarchy of stages. These stages of the inductive definition are indexed using some suitable notion of “ordinal.” The paradigm for a direct understanding of an inductive definition is that for the collection of natural numbers, which is characterized as the smallest collection containing zero and closed under the successor function.

Aspects of general topology in constructive set theory

Abstract Working in constructive set theory we formulate notions of constructive topological space and set-generated locale so as to get a good constructive general version of the classical Galois adjunction between topological spaces and locales. Our notion of constructive topological space allows for the space to have a class of points that need not be a set. Also our notion of locale allows the locale to have a class of elements that need not be a set. Class sized mathematical structures need to be allowed for in constructive set theory because the powerset axiom and the full separation scheme are necessarily missing from constructive set theory. We also consider the notion of a formal topology, usually treated in Intuitionistic type theory, and show that the category of set-generated locales is equivalent to the category of formal topologies. We exploit ideas of Palmgren and Curi to obtain versions of their results about when the class of formal points of a set-presentable formal topology form a set.

On Relating Type Theories and Set Theories

The original motivation1 for the work described in this paper was to determine the proof theoretic strength of the type theories implemented in the proof development systems Lego and Coq, [12],[4]. These type theories combine the impredicative type of propositions2, from the calculus of constructions, [5], with the inductive types and hierarchy of type universes of Martin-Lof’s constructive type theory, [13]. Intuitively there is an easy way to determine an upper bound on the proof theoretic strength. This is to use the ‘obvious’ types-as-sets interpretation of these type theories in a strong enough classical axiomatic set theory. The elementary forms of type of Martin-Lof’s type theory have their familiar set theoretic interpretation, the impredicative type of propositions can be interpreted as a two element set and the hierarchy of type universes can be interpreted using a corresponding hierarchy of strongly inaccessible cardinal numbers. The assumption of the existence of these cardinal numbers goes beyond the proof theoretic strength of ZFC. But Martin-Lof’s type theory, even with its W types and its hierarchy of universes, is not fully impredicative and has proof theoretic strength way below that of second order arithmetic. So it is not clear that the strongly inaccessible cardinals used in our upper bound are really needed. Of course the impredicative type of propositions does give a fully impredicative type theory, which certainly pushes up the proof theoretic strength to a set theory3, Z−, whose strength is well above that of second order arithmetic. The hierarchy of type universes will clearly lead to some further strengthening. But is it necessary to go beyond ZFC to get an upper bound?

Final Universes of Processes

We describe the final universe approach to the characterisation of semantic universes and illustrate it by giving characterisations of the universes of CCS and CSP processes.