Wim Veldman

Radboud University | RU · Faculty of Science

In the context of a weak formal theory called Basic Intuitionistic Mathematics BIM\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textsf{BIM}$$\end{document}, we study...

The paper discusses the place of intuitionistic mathematics within the whole of constructive mathematics. We discuss some of the principles Brouwer proposed and used, but were not accepted by Bishop, like the continuity principle, the fan theorem and the bar theorem. We show some of their consequences in constructive mathematics, like the Borel hie...

We show that the intuitionistic first-order theory of equality has continuum many complete extensions. We also study the Vitali equivalence relation and show there are many intuitionistically precise versions of it.

The paper is an introduction to intuitionistic mathematics

This paper appeared in: A. Rezus (ed.), Contemporary Logic and Computing, Series: Landscapes in Logic, Volume 1, College Publications, London, 2020, pp. 355-396.

We show that the intuitionistic first-order theory of equality has continuum many complete extensions. We also study the Vitali equivalence relation and show there are many intuitionistically precise versions of it.

This book collects fifteen essays. Thirteen of them were published in the period 1979–2002, and essays 14 and 15 appear now for the first time. The first essay was written together with Pierdaniele Giaretta, and essays 9, 11, and 14 are joint work with Gabriele Usberti. The essays treat a number of topics in the philosophy of mathematics, mainly in...

In the context of a weak formal theory called Basic Intuitionistic Mathematics BIM, we study Brouwer's Fan Theorem and a strong negation of the Fan Theorem, Kleene's Alternative (to the Fan Theorem). We prove that the Fan Theorem is equivalent to contrapositions of a number of intuition-istically accepted axioms of countable choice and that Kleene'...

Describing Brouwer's influence upon Wittgenstein's thinking

IN MEMORIAM: WALTER (WOUTER) VAN STIGT (1927–2015) - Volume 23 Issue 1 - Wim Veldman

We prove intuitionistic versions of the classical theorems saying that all countable closed subsets of [-π,π] and even all countable subsets of [-π,π] are sets of uniqueness. We introduce the co-derivative extension of an open subset of the set R of the real numbers as a constructively possibly more useful notion than the derivative of its compleme...

Est enim per contrapositionem conversio ut si dicas omnis homo animal est omne non animal non homo est. Boethius, de Syll. Cat. Abstract. In the earlier papers [31], [28] and [32] we collected statements that are, in a weak formal context, equivalent to Brouwer's Fan Theorem. This time, we do the same for the Principle of Open Induction on Cantor s...

As in the earlier paper [33], we collect statements that, in a weak formal theory called Basic Intuitionistic Mathematics BIM, are equivalent either to Brouwer's Fan Theorem or to Kleene's Alternative (to the Fan Theorem), a strong denial of the Fan Theorem. We prove, among other things, that the Fan Theorem is equivalent to a (positive) contraposi...

In a weak formal context, called BIM, for Basic Intuitionistic Mathematics, Brouwer's Fan Theorem is proven to be equivalent to a contraposition of a restricted axiom of countable binary choice, to the principle: positively inseparable enumerable subsets of IN intersect, to a contraposition of a restricted axiom of countable compact choice, to the...

The paper is a contribution to intuitionistic reverse mathematics. We introduce a formal system called Basic Intuitionistic Mathematics BIM, and then search for statements that are, over BIM, equivalent to Brouwer’s Fan Theorem or to its positive denial, Kleene’s Alternative to the Fan Theorem. The Fan Theorem is true under the intended intuitionis...

We study projective subsets of Baire space from Brouwer's intuitionistic point of view, using his Thesis on Bars and his continuity axioms. We first study analytic sets; these are the projections of the closed subsets of Baire space. We consider a number of examples and discover a fine structure in the class of the analytic sets that fail to be pos...

In intuitionistic analysis, a subset of a Polish space like R or N is called positively Borel if and only if it is an open subset of the space or a closed subset of the space or the result of forming either the countable union or the countable intersection of an infinite sequence of (earlier constructed) positively Borel subsets of the space. The o...

In a weak system for intuitionistic analysis, one may prove, using the Fan Theorem as an additional axiom, that, for every continuous function ø from the unit square U to itself, for every positive rational e, there exists x in U such that |ø(x) − x| < e. Conversely, if this statement is taken as an additional axiom, the Fan Theorem follows.

Taking Brouwer's intuitionistic standpoint, we examine finite and infinite games of perfect information for players I and II. If one understands the disjunction occurring in the classical notion of determinacy constructively, even finite games are not always determinate. We therefore suggest an intuitionistically different notion of determinacy and...

In intuitionistic analysis, Brouwer’s Continuity Principle implies, together with an Axiom of Countable Choice, that the positively Borel sets form a genuinely growing hierarchy: every level of the hierarchy contains sets that do not occur at any lower level.

One may distinguish two components in Brouwer’s proposal for an intuitionistic reform of the existing mathematical practice.

Without Abstract

L.E.J. Brouwer made a mistake in the formulation of his famous bar theorem, as was pointed out by S.C. Kleene. By repeating this mistake several times, Brouwer has caused confusion. We consider the assumption underlying his bar theorem, calling it Brouwer’s Thesis. This assumption is not refuted by Kleene’s example and we use it to obtain a conclus...

We study projective subsets of Baire space N from Brouwer's in-tuitionistic point of view, using his Thesis on Bars and his continuity axioms. Note that, also intuitionistically, N is homeomorphic to N × N. A subset of N is projective if it results from a closed or an open subset of N × N = N by a finite number of applications of the two operations...

In the context of intuitionistic real analysis, we introduce the set F con- sisting of all continuous functions φ from (0,1) to R such that φ(0) = 0 and φ(1) = 1. We let I0 be the set of all φ in F for which we may find x in (0,1) such that φ(x )= 1 2 .I t is well-known that there are functions in F that we can not prove to belong to I0 ,a nd that,...

The author proved in his Ph.D. Thesis [W. Veldman, Investigations in intuitionistic hierarchy theory, Ph.D. Thesis, Katholieke Universiteit Nijmegen, 1981] that, in intuitionistic analysis, the positively Borel subsets of Baire space N form a genuinely growing hierarchy: every level of the hierarchy contains sets that do not occur at any lower leve...

This paper is a slightly revised translation of [20] We calculate n such that, for every y in [0, 1], if 2 m+1 . Finally we find i such that i > n and 2 n+1 . We conclude: |x-y 2 n , and therefore: i )-f(y i )| < -f(x))| |f(y i )-f(x)| < 2 m . Contradiction. There must exist a suitable n

We study projective subsets of Baire space N from Brouwer's in-tuitionistic point of view, using his Thesis on Bars and his continuity axioms. Note that, also intuitionistically, N is homeomorphic to N × N. A subset of N is projective if it results from a closed or an open subset of N × N = N by a finite number of applications of the two operations...

We want to show, in this paper, that, in intuitionistic analysis, the union of two closed subsets of Baire space N is not always closed, and that, more generally, the union of a closed set and a II n 0-set is not always II n+10. In the proof of this fact we make use of the intuitionistic Borel Hierarchy Theorem, established in (Veldman, 1981) and (...

this paper is to show that the arguments given by Higman and Kruskal are essentially constructive and acceptable from an intuitionistic point of view and that the later argument given by Nash-Williams is not. The paper consists of the following 11 Sections. 1. Dickson's Lemma 2. Almost full relations 3. Brouwer's Thesis 4. Ramsey's Theorem 5. The F...

Brouwer's Continuity Principle distinguishes intuitionistic mathematics from other varieties of constructive mathematics, giving it its own avour. We discuss the plausibility of this assumption and show how it is used. We explain how one may understand its consequences even if one hesitates to accept it as an axiom. 1 Brouwer's Continuity Principle...

Consider the following converse of the Mean Value Theorem.Let f be a differentiable function on [a, b]. Ifcϵ (a, b), then there are α and β in [a, b] such that(f(β) − f(α))(β − α) = f′(c).Assuming some weak conditions to be mentioned in Section 3, Tong and Braza [3] were able to prove this statement. Unfortunately their proof does not provide a met...

In intuitionistic analysis one may prove, using Brouwer's continuity principle and an axiom of countable choice, that the positively Borel sets form a really growing hierarchy. The continuity principle implies also that the Borel hierarchy has a remarkable ne structure.

The double negation of a proposition is, in intuitionistic logic, many a time a weaker statement than the proposition itself, and frequently, assertions intermediate in strength between the two of them are easy to find. Propositions that coincide with their own double negation have been called stable, see van Dantzig (1947).

We establish constructive refinements of several well-known theorems in elementary model theory. The additive group of the real numbers may be embedded elementarily into the additive group of pairs of real numbers, constructively as well as classically.

At first sight, the argument which F.P. Ramsey gave for (the infinite case of) his famous theorem from 1927, is hopelessly unconstructive. If suitably reformulated, the theorem is true intuitionistically as well as classically: we offer a proof which should convince both the classical and the intuitionistic reader.

In descriptive set theory (cf. Moschovakis 1980), a subject which was founded in the early decades of this century by French and Russian mathematicians like Baire, Borel, Lebesgue, Lusin and Suslin, one describes and studies classes of subsets of the set IR of real numbers. Examples of such classes are: the class of open subsets of IR, the class of...

GabbayDov M.. Semantical investigations in Heyting's intuitionistic logic. Synthese library, vol. 148. D. Reidel Publishing Company, Dordrecht, Boston, and London, 1981, x + 287 pp. - Volume 51 Issue 3 - Wim Veldman

Publisher Summary This chapter describes two principles of reasoning—(i) for all subsets A of ωxω and (ii) for all subsets A of ωx ω ω—and shows that the principle (i), together with commonly accepted principles of intuitionistic analysis, leads to a contradiction. The principles (i) and (ii) simplify the task of classifying definable subsets of ω...

Investigations in Intuitionistic Hierarchy Theory Wim Veldman This thesis is concerned with constructive reasoning in descriptive set theory. The venerable subject of descriptive set theory was developed in the early decades of this century, mainly by French and Russian mathematicians. It started from the following observation: once the class of...

Although Brouwer became famous for his vehement attacks upon classical logic and set theory, his work did not develop in a vacuum and strongly depended on that of Cantor. His mind bent on shifting aside nonconstructive arguments, he tried to rebuild Cantor's edifice along new, intuitionistic lines. The continuum hypothesis, lying at the core of set...

The problem of treating the semantics of intuitionistic logic within the framework of intuitionistic mathematics was first attacked by E. W. Beth [1]. However, the completeness theorem he thought to have obtained, was not true, as was shown in detail in a report by V. H. Dyson and G. Kreisel [2]. Some vague remarks of Beth's, for instance in his bo...