A. S. Troelstra

Aspects of Constructive Mathematics

Publisher Summary In this chapter, constructive is meant as finitism, constructive recursive analysis, and intuitionism. The chapter discusses constructivism as the study of a special area in the whole of mathematical experience. The principal aspects of constructivism discussed in the chapter are the role of logic and abstract concepts; reductions to quantifier-free statements; and interpretation of the logical operations; intensional aspects; the validity of Churchs thesis; continuity axioms, the possibility of a theory of continuous; usefulness of the subjectivistic interpretation; and the quest for explicit definability. The chapter also discusses Markovs schema; connection between validity for intuitionistic predicate logic and mathematical assumptions. It presents the existence of classical counterparts to problems of constructive mathematics and systematic procedures for constructivizing classical theorems.

Note on the Fan Theorem

The principal aim of this paper is to establish a theorem stating, roughly, that the addition of the fan theorem and the. continuity schema to an intuitionistic system of elementary analysis results in a conservative extension with respect to arithmetical statements. The result implies that the consistency of first order arithmetic cannot be proved by use of the fan theorem, in addition to standard elementary methods—although it was the opposite assumption which led Gentzen to withdraw the first version of his consistency proof for arithmetic (see [B]). We must presuppose acquaintance with notation and principal results of [K, T], and with §1.6, Chapter II, and Chapter III, §4-6 of [T1]. In one respect we shall deviate from the notation in [K, T]: We shall use (n) x (instead of g(n, x)) to indicate the x th component of the sequence coded by n, if x n ), 0 otherwise. We also introduce abbreviations n ≤* m , a ≤ b which will be used frequently below:

On the Early History of Intuitionistic Logic

We describe the early history of intuitonistic logic, its formalization and the genesis of the so-called Brouwer-Heyting-Kolmogorov interpretation. In particular we discuss at some length whether Heyting’s papers contain an anticipation of logic with existence predicate. Finally we publish some source material, in particular letters of Bemays, Glivenko and Kolmogorov to Heyting.

Notions of Realizability for Intuitionistic Arithmetic and Intuitionistic Arithmetic in all Finite Types

Publisher Summary This chapter discusses properties of realizability and modified realizability interpretations for intuitionistic arithmetic HA and intuitionistic arithmetic in all finite types HA ω . The chapter describes the formal systems and presents the model hereditarily recursive operations (HRO) and hereditarily effective operations (HEO) for the intensional and the extensional version of HA ω respectively. The chapter characterizes the axiomatically formulae that can be proved in HA, resp. HA ω to be realizable, resp. modified realizable, resp. Dialectica interpretable. The chapter uses these results and the models HRO, HEO for conservative extension results and consistency results (e.g., consistency of HA with Markovs schema and Churchs thesis, HAW is conservative over HA, consistency of certain “axioms of choice” for HRO, HEO) and proves theoretic-closure conditions.

From constructivism to computer science

My field is mathematical logic, with a special interest in constructivism, and I would not dare to call myself a computer scientist. But some computer scientists regard my work as a contribution to their field; and in this text I shall try to explain how this is possible, by taking a look at the history of ideas. I want to describe how two interrelated ideas, connected with the constructivistic trend in the foundations of mathematics, developed within mathematical logic and ultimately diffused into computer science. It will be seen that this development has not been a quite straightforward one. In the history of ideas it often looks as if a certain idea has to be discovered several times, by different people, before it really enters into the consciousness of science.

On the Origin and Development of Brouwer's Concept of Choice Sequence

Publisher Summary This chapter discusses the development of the notion of choice sequences in Brouwers (published and unpublished) work. There seems to have been two basic motives for the introduction of choice sequences by Brouwer: to do justice to the intuitive content of the idea of continuum and to provide a basis for the theorem that a function on a bounded closed interval has to be uniformly continuous. Whether the intuitive content of the continuum has been caught by the notion of a choice sequence will remain to some extent a matter of personal insight and feeling. On the one hand, Brouwers later theory of the continuum seems far more coherent and successful in catching the underlying intuition, while on the other hand, further development of the analytic point of view has shown that the intuition of the continuum is much more complicated than is suggested by Brouwers writings. Thus, perhaps choice sequences have not yielded the harvest Brouwer expected, but their study has been rewarding in ways not foreseen by Brouwer and will remain of interest to the philosophy of mathematics.

The Theory of Choice Sequences

Publisher Summary This chapter describes a formal system for intuitionistic analysis and the intuitive justification of this system. The system discussed is called “Choice Sequence” (CS). CS contains numerical variables, two kinds of variables for constructive functions, and variables for choice sequences. The subsystem of CS obtained by restricting the language, axioms, and the rules of CS to expressions not involving variables for choice sequences is called IDK, because the main feature of the subsystem is an inductive definition of a class of constructive functions called K. The formal work on the system CS was started by Kreisel (which was privately circulated only) and was recently improved and rounded off to a certain extent by a joint effort of Kreisel and the author.

A note on non-extensional operations in connection with continuity and recursiveness

Abstract It is shown that relative to intuitionistic arithmetic in all finite types extensionality and axioms of choice are incompatible with Churchs thesis or continuity axioms; these results are contrasted with consistency results for cases where less choice or extensionality is assumed.

Realizability and intuitionistic logic

The present paper is an attempt to clarify the relationship between Heyting’s well-known explanation of the intuitionistic logical operators on the one hand, and realizability interpretations on the other hand, in particular in connection with the theory of types as developed by P. Martin-Lof. Part of the discussion may be regarded as a supplement to the discussion by Martin-Lof.

Projections of Lawless Sequences

Publisher Summary This chapter describes the projections of lawless sequences. Intuitionistically, a wide variety of notions of choice sequence may be conceived. Many of the notions are of little use to the construction of a theory of real numbers because of their anti-social behavior. They provide a testing ground for intuitionistic concepts and principles, and represent interesting examples of topological spaces. The chapter focusses on the anti-social notions of choice sequence. The anti-social behavior of the notions considered is due to the fact that the restricting conditions do not contain non-lawlike parameters in these cases. The chapter discusses some of the notions by sequences that are obtained by certain lawlike transformations (called projection operators) on lawless sequences (of lawlike objects). The resulting sequences are called projected sequences. Projected sequences may serve the same mathematical purposes as the notions because the projected sequences are approximations of certain notions. Thus, the number of notions required in intuitionistic mathematics is reduced. The chapter describes a modified notion that is basic and is closely related to Myhills notion.