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Author : John Stillwell
ISBN : 9780691196411
Genre : Mathematics
File Size : 39.46 MB
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" This book presents reverse mathematics to a general mathematical audience for the first time. Reverse mathematics is a new field that answers some old questions. In the two thousand years that mathematicians have been deriving theorems from axioms, it has often been asked: which axioms are needed to prove a given theorem? Only in the last two hundred years have some of these questions been answered, and only in the last forty years has a systematic approach been developed. In Reverse Mathematics, John Stillwell gives a representative view of this field, emphasizing basic analysis--finding the "right axioms" to prove fundamental theorems--and giving a novel approach to logic. Stillwell introduces reverse mathematics historically, describing the two developments that made reverse mathematics possible, both involving the idea of arithmetization. The first was the nineteenth-century project of arithmetizing analysis, which aimed to define all concepts of analysis in terms of natural numbers and sets of natural numbers. The second was the twentieth-century arithmetization of logic and computation. Thus arithmetic in some sense underlies analysis, logic, and computation. Reverse mathematics exploits this insight by viewing analysis as arithmetic extended by axioms about the existence of infinite sets. Remarkably, only a small number of axioms are needed for reverse mathematics, and, for each basic theorem of analysis, Stillwell finds the "right axiom" to prove it. By using a minimum of mathematical logic in a well-motivated way, Reverse Mathematics will engage advanced undergraduates and all mathematicians interested in the foundations of mathematics. "--

Author : Stephen G. Simpson
ISBN : 9781108637220
Genre : Mathematics
File Size : 33.1 MB
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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.

Author : Denis R Hirschfeldt
ISBN : 9789814612630
Genre : Mathematics
File Size : 90.40 MB
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This book is a brief and focused introduction to the reverse mathematics and computability theory of combinatorial principles, an area of research which has seen a particular surge of activity in the last few years. It provides an overview of some fundamental ideas and techniques, and enough context to make it possible for students with at least a basic knowledge of computability theory and proof theory to appreciate the exciting advances currently happening in the area, and perhaps make contributions of their own. It adopts a case-study approach, using the study of versions of Ramsey's Theorem (for colorings of tuples of natural numbers) and related principles as illustrations of various aspects of computability theoretic and reverse mathematical analysis. This book contains many exercises and open questions. Contents:Setting Off: An IntroductionGathering Our Tools: Basic Concepts and NotationFinding Our Path: König's Lemma and ComputabilityGauging Our Strength: Reverse MathematicsIn Defense of DisarrayAchieving Consensus: Ramsey's TheoremPreserving Our Power: ConservativityDrawing a Map: Five DiagramsExploring Our Surroundings: The World Below RT22Charging Ahead: Further TopicsLagniappe: A Proof of Liu's Theorem Readership: Graduates and researchers in mathematical logic. Key Features:This book assumes minimal background in mathematical logic and takes the reader all the way to current research in a highly active areaIt is the first detailed introduction to this particular approach to this area of researchThe combination of fully worked out arguments and exercises make this book well suited to self-study by graduate students and other researchers unfamiliar with the areaKeywords:Reverse Mathematics;Computability Theory;Computable Mathematics;Computable Combinatorics

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 : 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 : Nicholas Henry
ISBN : 1092277889
Genre :
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First we provide a unified framework for developing theories of Bounded Arithmetic that are associated with uniform classes inside polytime (P) in the same way that Buss's theory S 12 is associated with P. We obtain finitely axiomatized theories many of which turn out to be equivalent to a number of existing systems. By formalizing the proof of Barrington's Theorem (that the functions computable by polynomial-size bounded-width branching programs are precisely functions computable in ALogTime, or equivalently NC 1 ) we prove one such equivalence between the theories associated with ALogTime, solving a problem that remains open in [Ara00, Pit00]. Our theories demonstrate an advantage of the simplicity of Zambella's two-sorted setting for small theories of Bounded Arithmetic. Then we give the first definitions for the relativizations of small classes such as NC 1 , L, NL that preserve their inclusion order. Separating these relativized classes is shown to be as hard as separating the corresponding non-relativized classes. Our framework also allows us to obtain relativized theories that characterize the newly defined relativized classes. Finally we formalize and prove a number of mathematical theorems in our theories. In particular, we prove the discrete versions of the Jordan Curve Theorem in the theories V 0 and V 0 (2), and establish some facts about the distribution of prime numbers in the theory VTC 0 . Our V 0 - and V 0 (2)-proofs improve a number of existing upper bounds for the propositional complexity of combinatorial principles related to grid graphs. Overall, this thesis is a contribution to Bounded Reverse Mathematics, a theme iiwhose purpose is to formalize and prove (discrete versions of) mathematical theorems in the weakest possible theories of bounded arithmetic.

"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.

Author : John Stillwell
ISBN : 9780691178547
Genre : Mathematics
File Size : 81.77 MB
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Elements of Mathematics takes readers on a fascinating tour that begins in elementary mathematics—but, as John Stillwell shows, this subject is not as elementary or straightforward as one might think. Not all topics that are part of today's elementary mathematics were always considered as such, and great mathematical advances and discoveries had to occur in order for certain subjects to become "elementary." Stillwell examines elementary mathematics from a distinctive twenty-first-century viewpoint and describes not only the beauty and scope of the discipline, but also its limits. From Gaussian integers to propositional logic, Stillwell delves into arithmetic, computation, algebra, geometry, calculus, combinatorics, probability, and logic. He discusses how each area ties into more advanced topics to build mathematics as a whole. Through a rich collection of basic principles, vivid examples, and interesting problems, Stillwell demonstrates that elementary mathematics becomes advanced with the intervention of infinity. Infinity has been observed throughout mathematical history, but the recent development of "reverse mathematics" confirms that infinity is essential for proving well-known theorems, and helps to determine the nature, contours, and borders of elementary mathematics. Elements of Mathematics gives readers, from high school students to professional mathematicians, the highlights of elementary mathematics and glimpses of the parts of math beyond its boundaries.

Author : Roman Murawski
ISBN : 9789401728669
Genre : Philosophy
File Size : 67.90 MB
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Recursive Functions and Metamathematics deals with problems of the completeness and decidability of theories, using as its main tool the theory of recursive functions. This theory is first introduced and discussed. Then Gödel's incompleteness theorems are presented, together with generalizations, strengthenings, and the decidability theory. The book also considers the historical and philosophical context of these issues and their philosophical and methodological consequences. Recent results and trends have been included, such as undecidable sentences of mathematical content, reverse mathematics. All the main results are presented in detail. The book is self-contained and presupposes only some knowledge of elementary mathematical logic. There is an extensive bibliography. Readership: Scholars and advanced students of logic, mathematics, philosophy of science.

This is the first treatment in book format of proof-theoretic transformations - known as proof interpretations - that focuses on applications to ordinary mathematics. It covers both the necessary logical machinery behind the proof interpretations that are used in recent applications as well as – via extended case studies – carrying out some of these applications in full detail. This subject has historical roots in the 1950s. This book for the first time tells the whole story.

Author : Barry S. Cooper
ISBN : 9783642308703
Genre : Computers
File Size : 34.29 MB
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This book constitutes the refereed proceedings of the Turing Centenary Conference and the 8th Conference on Computability in Europe, CiE 2012, held in Cambridge, UK, in June 2012. The 53 revised papers presented together with 6 invited lectures were carefully reviewed and selected with an acceptance rate of under 29,8%. The CiE 2012 Turing Centenary Conference will be remembered as a historic event in the continuing development of the powerful explanatory role of computability across a wide spectrum of research areas. The papers presented at CiE 2012 represent the best of current research in the area, and forms a fitting tribute to the short but brilliant trajectory of Alan Mathison Turing. Both the conference series and the association promote the development of computability-related science, ranging over mathematics, computer science and applications in various natural and engineering sciences such as physics and biology, and also including the promotion of related non-scientific fields such as philosophy and history of computing.