Constructive reverse mathematics : Habilitationsschrift

We give a comprehensive treatment of the area known as constructive reverse mathematics, which aims to classify, over intuitionistic logic, various theorems in intuitionistic, constructive recursive, and classical mathematics. Our work provides an overview of many known results, but also many new results, and new proofs of known results. We cover omniscience principles, Markov’s principle, Brouwer’s fan theorem(s), recursive principles, Ishihara’s boundedness principle BD-N, as well as minor principles, and separation techniques.Diese Arbeit beschäftigt sich mit dem Gebiet der „konstruktiven reversen Mathematik“ („constructive reverse mathematics“), dessen Ziel es ist verschiedene Aussagen aus der intuitionistischen, der rekursiven und der klassischen Mathematik über intuitionistischer Logik zu klassifizieren. Es wird ein Überblick über bekannte Ergebnisse, aber auch viele neue Resultate und neue Beweise gegeben. Unter anderem werden „Allwissenheits“-Prinzipien, Markov’s Prinzip, die Brouwer’schen Fächersätze, Aussagen aus der rekursiven Mathematik und Ishiharas Beschränktheitsprinzip BD-N behandelt

The Third Trick

We prove a result, similar to the ones known as Ishihara's First and Second Trick, for sequences of functions

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

The Third Trick

We prove a result, similar to the ones known as Ishihara's First and Second Trick, for sequences of functions