Constructive mathematics – mathematics in which 'there exists' always means 'we can construct' – is enjoying a renaissance. fifty years on from Bishop's groundbreaking account of constructive analysis, constructive mathematics has spread out to touch almost all areas of mathematics and to have profound influence in theoretical computer science. This handbook gives the most complete overview of modern constructive mathematics, with contributions from leading specialists surveying the subject's myriad aspects. Major themes include: constructive algebra and geometry, constructive analysis, constructive topology, constructive logic and foundations of mathematics, and computational aspects of constructive mathematics. A series of introductory chapters provides graduate students and other newcomers to the subject with foundations for the surveys that follow. Edited by four of the most eminent experts in the field, this is an indispensable reference for constructive mathematicians and a fascinating vista of modern constructivism for the increasing number of researchers interested in constructive approaches.

The overall topic of the volume, Mathematics for Computation (M4C), is mathematics taking crucially into account the aspect of computation, investigating the interaction of mathematics with computation, bridging the gap between mathematics and computation wherever desirable and possible, and otherwise explaining why not.Recently, abstract mathematics has proved to have more computational content than ever expected. Indeed, the axiomatic method, originally intended to do away with concrete computations, seems to suit surprisingly well the programs-from-proofs paradigm, with abstraction helping not only clarity but also efficiency.Unlike computational mathematics, which rather focusses on objects of computational nature such as algorithms, the scope of M4C generally encompasses all the mathematics, including abstract concepts such as functions. The purpose of M4C actually is a strongly theory-based and therefore, is a more reliable and sustainable approach to actual computation, up to the systematic development of verified software.While M4C is situated within mathematical logic and the related area of theoretical computer science, in principle it involves all branches of mathematics, especially those which prompt computational considerations. In traditional terms, the topics of M4C include proof theory, constructive mathematics, complexity theory, reverse mathematics, type theory, category theory and domain theory.The aim of this volume is to provide a point of reference by presenting up-to-date contributions by some of the most active scholars in each field. A variety of approaches and techniques are represented to give as wide a view as possible and promote cross-fertilization between different styles and traditions.

This book on proof theory centers around the legacy of Kurt Schutte and its current impact on the subject. Schutte was the last doctoral student of David Hilbert who was the first to see that proofs can be viewed as structured mathematical objects amenable to investigation by mathematical methods (metamathematics). Schutte inaugurated the important paradigm shift from finite proofs to infinite proofs and developed the mathematical tools for their analysis. Infinitary proof theory flourished in his hands in the 1960s, culminating in the famous bound 0 for the limit of predicative mathematics (a fame shared with Feferman). Later his interests shifted to developing infinite proof calculi for impredicative theories. Schutte had a keen interest in advancing ordinal analysis to ever stronger theories and was still working on some of the strongest systems in his eighties. The articles in this volume from leading experts close to his research, show the enduring influence of his work in modern proof theory. They range from eye witness accounts of his scientific life to developments at the current research frontier, including papers by Schutte himself that have never been published before.

Gerhard Gentzen has been described as logic's lost genius, whom Goedel called a better logician than himself. This work comprises articles by leading proof theorists, attesting to Gentzen's enduring legacy to mathematical logic and beyond. The contributions range from philosophical reflections and re-evaluations of Gentzen's original consistency proofs to the most recent developments in proof theory. Gentzen founded modern proof theory. His sequent calculus and natural deduction system beautifully explain the deep symmetries of logic. They underlie modern developments in computer science such as automated theorem proving and type theory.

This book on proof theory centers around the legacy of Kurt Schutte and its current impact on the subject. Schutte was the last doctoral student of David Hilbert who was the first to see that proofs can be viewed as structured mathematical objects amenable to investigation by mathematical methods (metamathematics). Schutte inaugurated the important paradigm shift from finite proofs to infinite proofs and developed the mathematical tools for their analysis. Infinitary proof theory flourished in his hands in the 1960s, culminating in the famous bound 0 for the limit of predicative mathematics (a fame shared with Feferman). Later his interests shifted to developing infinite proof calculi for impredicative theories. Schutte had a keen interest in advancing ordinal analysis to ever stronger theories and was still working on some of the strongest systems in his eighties. The articles in this volume from leading experts close to his research, show the enduring influence of his work in modern proof theory. They range from eye witness accounts of his scientific life to developments at the current research frontier, including papers by Schutte himself that have never been published before.

Diplomarbeit aus dem Jahr 2000 im Fachbereich BWL - Personal und Organisation, Note: 2,0, Hochschule Bremen (Wirtschaft), Sprache: Deutsch, Abstract: Inhaltsangabe: Gang der Untersuchung: In der vorliegenden Arbeit geht es um die Frage, wie die Staatstatigkeit effektiver gestaltet werden kann. Zunachst erfolgt eine volkswirtschaftliche Einfuhrung in das Thema, in der die Notwendigkeit eines staatlichen Eingreifens in den Wettbewerb untersucht wird. Staatliches Eingreifen kann erforderlich sein, wenn es durch okonomisches Handeln zu externen Effekten kommt. In diesen Fallen sind unbeteiligte Dritte vom okonomischen Handeln zweier Wirtschaftsparteien betroffen. Es erfolgt eine Untersuchung von positiven und negativen externen Effekten. Da die Verursacher externer Effekte kein okonomisches Interesse haben, die Externalitaten zu optimieren, mussen Methoden gefunden werden, diese zu internalisieren. Im ersten Teil dieser Arbeit werden diese Internalisierungsverfahren beschrieben. Es wird untersucht, warum diese Verfahren scheitern konnen. In diesen Fallen des Marktversagens muss der Staat eingreifen. Der Staat greift hier in den Wettbewerb ein, indem er offentliche Guter und Leistungen zur Verfugung stellt bzw. Gesetze und Regelungen erlasst. Aufgrund seiner Organisation ist es ihm oftmals nicht moglich, dies in effizienter Weise zu tun. Es werden die Ursachen dieses Staatsversagens untersucht und im weiteren Moglichkeiten diskutiert, die Ausfuhrung und Organisation der staatlichen Tatigkeit effizienter zu gestalten. Bereits Max Weber hat Anfang des letzten Jahrhunderts das Staatsversagen erkannt. Er hat in seinen Schriften und Reden angesprochen, dass es aufgrund einer Mentalitat der allgemeinen Rationalisierung zu einer Uberbetonung der staatlichen Verwaltung und somit zu einer Beamtenherrschaft kommen kann. Weber forderte schon zu Beginn des 20. Jahrhunderts eine starkere Kontrolle des Beamtentums, da es naturgemass nicht im Interesse der Beamten lage, im Sinne der Of