Hannes Diener

Sequences of real functions on [0,1] in constructive reverse mathematics

Abstract We give an overview of the role of equicontinuity of sequences of real-valued functions on [ 0 , 1 ] and related notions in classical mathematics, intuitionistic mathematics, Bishop’s constructive mathematics, and Russian recursive mathematics. We then study the logical strength of theorems concerning these notions within the programme of Constructive Reverse Mathematics. It appears that many of these theorems, like a version of Ascoli’s Lemma, are equivalent to fan-theoretic principles.

The anti-Specker property, positivity, and total boundedness

Working within Bishop-style constructive mathematics, we examine some of the consequences of the anti-Specker property, known to be equivalent to a version of Brouwers fan theorem. The work is a contribution to constructive reverse mathematics (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

SEPARATING THE FAN THEOREM AND ITS WEAKENINGS

Varieties of the Fan Theorem have recently been developed in reverse constructive mathematics, corresponding to different continuity principles. They form a natural implicational hierarchy. Some of the implications have been shown to be strict, others strict in a weak context, and yet others not at all, using disparate techniques. Here we present a family of related Kripke models which separates all of the as yet identified fan theorems.

A constructive treatment of Urysohn's Lemma in an apartness space

This paper is dedicated to Prof. Dr. Gunter Asser, whose work in founding this journal and maintaining it over many difficult years has been a major contribution to the activities of the mathematical logic community. At first (maybe even at second) sight it appears highly unlikely that Urysohns Lemma has any significant constructive content. However, working in the context of an apartness space and using functions whose values are a generalisation of the reals, rather than real numbers, enables us to produce a significant constructive version of that lemma. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Separating the Fan Theorem and Its Weakenings

Varieties of the Fan Theorem have recently been developed in reverse constructive mathematics, corresponding to different continuity principles. They form a natural implicational hierarchy. Some of the implications have been shown to be strict, others strict in a weak context, and yet others not at all, using disparate techniques. Here we present a family of related Kripke models which suffices to separate all of the as yet identified fan theorems.

Reclassifying the antithesis of Specker's theorem

It is shown that a principle, which can be seen as a constructivised version of sequential compactness, is equivalent to a form of Brouwer’s fan theorem. The complexity of the latter depends on the geometry of the spaces involved in the former.

Logic, Construction, Computation

Over the last few decades the interest of logicians and mathematicians in constructive and computational aspects of their subjects has been steadily growing, and researchers from disparate areas realized that they can benefit enormously from the mutual exchange of techniques concerned with those aspects. A key figure in this exciting development is the logician and mathematician Helmut Schwichtenberg to whom this volume is dedicated on the occasion of his 70th birthday and his turning emeritus. The volume contains 20 articles from leading experts about recent developments in Constructive set theory, Provably recursive functions, Program extraction, Theories of truth, Constructive mathematics, Classical versus intuitionistic logic, Inductive definitions, and Continuous functionals and domains.

Uniqueness, Continuity, and Existence of Implicit Functions in Constructive Analysis.

We extract a quantitative variant of uniqueness from the usual hypotheses of the implicit functions theorem. This leads not only to an a priori proof of continuity, but also to an alternative, fully constructive existence proof.

Notions of Cauchyness and Metastability

We show that several weakenings of the Cauchy condition are all equivalent under the assumption of countable choice, and investigate to what extent choice is necessary. We also show that the syntactically reminiscent notion of metastability allows similar variations, but is empty in terms of its constructive content.