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Author : Stephen George Simpson
ISBN : 3540648828
Genre : Mathematics
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An original contribution to the foundations of mathematics, with emphasis on the role of set existence axioms, this book gives particular attention to several well known foundational programs including those by Hilbert, Bishop, and Weyl.

Author : Stephen G. Ross
ISBN : 9781439864289
Genre : Mathematics
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Reverse Mathematics is a program of research in the foundations of mathematics, motivated by the foundational questions of what are appropriate axioms for mathematics, and what are the logical strengths of particular axioms and particular theorems. The book contains 24 original papers by leading researchers. These articles exhibit the exciting rece

Author : Stephen G. Simpson
ISBN : 9781108637220
Genre : Mathematics
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Since their inception, the Perspectives in Logic and Lecture Notes in Logic series have published seminal works by leading logicians. Many of the original books in the series have been unavailable for years, but they are now in print once again. Reverse mathematics is a program of research in the foundations of mathematics, motivated by two foundational questions: 'what are appropriate axioms for mathematics?' and 'what are the logical strengths of particular axioms and particular theorems?' This volume, the twenty-first publication in the Lecture Notes in Logic series, contains twenty-four original research papers from respected authors that present exciting new developments in reverse mathematics and subsystems of second order arithmetic since 1998.

Gödel's true-but-unprovable sentence from the first incompleteness theorem is purely logical in nature, i.e. not mathematically natural or interesting. An interesting problem is to find mathematically natural and interesting statements that are similarly unprovable. A lot of research has since been done in this direction, most notably by Harvey Friedman. A lot of examples of concrete incompleteness with real mathematical content have been found to date. This brief contributes to Harvey Friedman's research program on concrete incompleteness for higher-order arithmetic and gives a specific example of concrete mathematical theorems which is expressible in second-order arithmetic but the minimal system in higher-order arithmetic to prove it is fourth-order arithmetic. This book first examines the following foundational question: are all theorems in classic mathematics expressible in second-order arithmetic provable in second-order arithmetic? The author gives a counterexample for this question and isolates this counterexample from the Martin-Harrington Theorem in set theory. It shows that the statement “Harrington's principle implies zero sharp" is not provable in second-order arithmetic. This book further examines what is the minimal system in higher-order arithmetic to prove the theorem “Harrington's principle implies zero sharp" and shows that it is neither provable in second-order arithmetic or third-order arithmetic, but provable in fourth-order arithmetic. The book also examines the large cardinal strength of Harrington's principle and its strengthening over second-order arithmetic and third-order arithmetic.

This volume discusses various aspects of Harvey Friedman's research in the foundations of mathematics over the past fifteen years. It should appeal to a wide audience of mathematicians, computer scientists, and mathematically oriented philosophers.

Author : Edward Craig
ISBN : 9781134344086
Genre : Philosophy
File Size : 64.82 MB
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The Shorter REP presents the very best of the acclaimed ten volume Routledge Encyclopedia of Philosophy in a single volume. It makes a selection of the most important entries available for the first time and covers all you need to know about philosophy, from Aristotle to Wittgenstein and animals and ethics to scientific method. Comprising over 900 entries and covering the major philosophers and philosophical topics, The Shorter REP includes the following special features: Unrivalled coverage of major philosophers, themes, movements and periods making the volume indispensable for any student or general reader Fully cross-referenced Revised versions of many of the most important entries, including fresh suggestions for further reading Over twenty brand new entries on important new topics such as Cloning and Sustainability entries by many leading philosophers such as Bernard Williams, Martha Nussbaum, Richard Rorty, Onora O'Neill, T.M. Scanlon and Anthony Appiah Striking new text design to help locate key entries quickly and easily An outstanding guide to all things philosophical, The Shorter Routledge Encyclopedia of Philosophy provides an unrivalled introduction to the subject for students and general readers alike.

"Foundations of the Formal Sciences" (FotFS) is a series of interdisciplinary conferences in mathematics, philosophy, computer science and linguistics. The main goal is to reestablish the traditionally strong links between these areas of research that have been lost in the past decades. The second conference in the series had the subtitle "Applications of Mathematical Logic in Philosophy and Linguistics" and brought speakers from all parts of the Formal Sciences together to give a holistic view of how mathematical methods can improve our philosophical and technical understanding of language and scientific discourse, ranging from the theoretical level up to applications in language recognition software. Audience: This volume is of interest to all formal philosophers and theoretical linguists. In addition to that, logicians interested in the applications of their field and logic students in mathematics, computer science, philosophy and linguistics can use the volume to broaden their knowledge of applications of logic.

This volume contains the proceedings of the Workshop on Logic and Computation, held in July 1987 at Carnegie-Mellon University. The focus of the workshop was the refined interaction between mathematics and computation theory, one of the most fascinating and potentially fruitful developments in logic. The importance of this interaction lies not only in the emergence of the computer as a powerful tool in mathematics research, but also in the various attempts to carry out significant parts of mathematics in computationally informative ways. The proceedings pursue three complementary aims: to develop parts of mathematics under minimal set-theoretic assumptions; to provide formal frameworks suitable for computer implementation; and to extract, from formal proofs, mathematical and computational information. Aimed at logicians, mathematicians, and computer scientists, this volume is rich in results and replete with mathematical, logical, and computational problems.

On the occasion of the retirement of Wolfram Pohlers the Institut für Mathematische Logik und Grundlagenforschung of the University of Münster organized a colloquium and a workshop which took place July 17 – 19, 2008. This event brought together proof theorists from many parts of the world who have been acting as teachers, students and collaborators of Wolfram Pohlers and who have been shaping the field of proof theory over the years. The present volume collects papers by the speakers of the colloquium and workshop; and they produce a documentation of the state of the art of contemporary proof theory.

Kurt Gödel (1906–1978) did groundbreaking work that transformed logic and other important aspects of our understanding of mathematics, especially his proof of the incompleteness of formalized arithmetic. This book on different aspects of his work and on subjects in which his ideas have contemporary resonance includes papers from a May 2006 symposium celebrating Gödel's centennial as well as papers from a 2004 symposium. Proof theory, set theory, philosophy of mathematics, and the editing of Gödel's writings are among the topics covered. Several chapters discuss his intellectual development and his relation to predecessors and contemporaries such as Hilbert, Carnap, and Herbrand. Others consider his views on justification in set theory in light of more recent work and contemporary echoes of his incompleteness theorems and the concept of constructible sets.

Author : Giovanni Sambin
ISBN : 9780191606939
Genre : Mathematics
File Size : 43.2 MB
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Per Martin-Löf's work on the development of constructive type theory has been of huge significance in the fields of logic and the foundations of mathematics. It is also of broader philosophical significance, and has important applications in areas such as computing science and linguistics. This volume draws together contributions from researchers whose work builds on the theory developed by Martin-Löf over the last twenty-five years. As well as celebrating the anniversary of the birth of the subject it covers many of the diverse fields which are now influenced by type theory. It is an invaluable record of areas of current activity, but also contains contributions from N. G. de Bruijn and William Tait, both important figures in the early development of the subject. Also published for the first time is one of Per Martin-Löf's earliest papers.

Author : Corrie E. Ingall
ISBN : OCLC:1164817039
Genre :
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Reverse Mathematics is a subfield of Computability Theory and mathematical logic concerned with one main question: "What are the necessary axioms for mathematics?" Reverse Mathematics allows us to characterize the logical strength of theorems, relating theorems across many mathematical disciplines. In this paper, we synthesize the historical work on the Reverse Math of Real Analysis by Simpson and Brown in a way that is accessible to non-logicians while a useful reference to those within the field. In particular, we describe the nuances of different definitions of closed sets, whether as complements of open sets or as sets that contain all their limit points. These distinct definitions connect to a relatively large gap of logical strength. In one case, we correct an error in the proof that "All closed sets are separably closed" implies ACA_0 over RCA_0 from Brown's 1990 paper "Notions of Closed Subsets of a Complete Separable Metric Space in Weak Subsystems of Second Order Arithmetic." We also consider how the dual definitions of closed sets affect the strength of the Baire Category Theorem. As a whole, this paper outlines a path of understanding Reverse Mathematics for those outside the field by exploring fundamental ideas from Real Analysis.

Author : Richard L. Epstein
ISBN : 9781400841554
Genre : Mathematics
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In Classical Mathematical Logic, Richard L. Epstein relates the systems of mathematical logic to their original motivations to formalize reasoning in mathematics. The book also shows how mathematical logic can be used to formalize particular systems of mathematics. It sets out the formalization not only of arithmetic, but also of group theory, field theory, and linear orderings. These lead to the formalization of the real numbers and Euclidean plane geometry. The scope and limitations of modern logic are made clear in these formalizations. The book provides detailed explanations of all proofs and the insights behind the proofs, as well as detailed and nontrivial examples and problems. The book has more than 550 exercises. It can be used in advanced undergraduate or graduate courses and for self-study and reference. Classical Mathematical Logic presents a unified treatment of material that until now has been available only by consulting many different books and research articles, written with various notation systems and axiomatizations.

Author : David Reed Solomon
ISBN : CORNELL:31924083817944
Genre :
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We study theorems of ordered group theory from the viewpoints of reverse mathematics and computable mathematics. Reverse mathematics uses subsystems of second order arithmetic to determine which set existence axioms are required to prove particular theorems. We give equivalences of $WKL\sb0$ (the orderability of direct products and nilpotent groups, the classical semigroup conditions for orderability), $ACA\sb0$ (the (existence of induced orders on quotient groups, the existence of the center of a group) and $\Pi\sbsp{1}{1} - CA\sb0$ (the classification of order types for countable fully ordered abelian groups). We show that $RCA\sb0$ suffices to prove Holder's Theorem and the theorem that every ordered group is the quotient of an ordered free group by a convex normal subgroup. In computable mathematics, we examine the complexity of orders of computable abelian and nilpotent groups and we study the relationship between spaces of orders of computable groups and computably bounded $\Pi\sbsp{1}{0}$ classes. Finally, we show that every orderable computable abelian group has a computable presentation with a computable order. This work answers several open questions from Downey and Kurtz (1986).

Author : William S. Levine
ISBN : 0849300533
Genre : Technology & Engineering
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Sifting through the variety of control systems applications can be a chore. Diverse and numerous technologies inspire applications ranging from float valves to microprocessors. Relevant to any system you might use, the highly adaptable Control System Fundamentals fills your need for a comprehensive treatment of the basic principles of control system engineering. This overview furnishes the underpinnings of modern control systems. Beginning with a review of the required mathematics, major subsections cover digital control and modeling. An international panel of experts discusses the specification of control systems, techniques for dealing with the most common and important control system nonlinearities, and digital implementation of control systems, with complete references. This framework yields a primary resource that is also capable of directing you to more detailed articles and books. This self-contained reference explores the universal aspects of control that you need for any application. Reliable, up-to-date, and versatile, Control System Fundamentals answers your basic control systems questions and acts as an ideal starting point for approaching any control problem.

This comprehensive monograph presents a detailed overview of creative works by the author and other 20th-century logicians that includes applications of proof theory to logic as well as other areas of mathematics. 1975 edition.