Almost Free Modules
Almost Free Modules

Author: P.C. Eklof

Publisher: Elsevier

Published: 1990-04-23

Total Pages: 498

ISBN-13: 0080960243

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This is an extended treatment of the set-theoretic techniques which have transformed the study of abelian group and module theory over the last 15 years. Part of the book is new work which does not appear elsewhere in any form. In addition, a large body of material which has appeared previously (in scattered and sometimes inaccessible journal articles) has been extensively reworked and in many cases given new and improved proofs. The set theory required is carefully developed with algebraists in mind, and the independence results are derived from explicitly stated axioms. The book contains exercises and a guide to the literature and is suitable for use in graduate courses or seminars, as well as being of interest to researchers in algebra and logic.

Extensions of the Axiom of Determinacy
Extensions of the Axiom of Determinacy

Author: Paul B. Larson

Publisher: American Mathematical Society

Published: 2023-10-19

Total Pages: 182

ISBN-13: 1470472104

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This is an expository account of work on strong forms of the Axiom of Determinacy (AD) by a group of set theorists in Southern California, in particular by W. Hugh Woodin. The first half of the book reviews necessary background material, including the Moschovakis Coding Lemma, the existence of strong partition cardinals, and the analysis of pointclasses in models of determinacy. The second half of the book introduces Woodin's axiom system $mathrm{AD}^{+}$ and presents his initial analysis of these axioms. These results include the consistency of $mathrm{AD}^{+}$ from the consistency of AD, and its local character and initial motivation. Proofs are given of fundamental results by Woodin, Martin, and Becker on the relationships among AD, $mathrm{AD}^{+}$, the Axiom of Real Determinacy, and the Suslin property. Many of these results are proved in print here for the first time. The book briefly discusses later work and fundamental questions which remain open. The study of models of $mathrm{AD}^{+}$ is an active area of contemporary research in set theory. The presentation is aimed at readers with a background in basic set theory, including forcing and ultrapowers. Some familiarity with classical results on regularity properties for sets of reals under AD is also expected.

Reflections on the Foundations of Mathematics
Reflections on the Foundations of Mathematics

Author: Stefania Centrone

Publisher: Springer Nature

Published: 2019-11-11

Total Pages: 511

ISBN-13: 3030156559

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This edited work presents contemporary mathematical practice in the foundational mathematical theories, in particular set theory and the univalent foundations. It shares the work of significant scholars across the disciplines of mathematics, philosophy and computer science. Readers will discover systematic thought on criteria for a suitable foundation in mathematics and philosophical reflections around the mathematical perspectives. The volume is divided into three sections, the first two of which focus on the two most prominent candidate theories for a foundation of mathematics. Readers may trace current research in set theory, which has widely been assumed to serve as a framework for foundational issues, as well as new material elaborating on the univalent foundations, considering an approach based on homotopy type theory (HoTT). The third section then builds on this and is centred on philosophical questions connected to the foundations of mathematics. Here, the authors contribute to discussions on foundational criteria with more general thoughts on the foundations of mathematics which are not connected to particular theories. This book shares the work of some of the most important scholars in the fields of set theory (S. Friedman), non-classical logic (G. Priest) and the philosophy of mathematics (P. Maddy). The reader will become aware of the advantages of each theory and objections to it as a foundation, following the latest and best work across the disciplines and it is therefore a valuable read for anyone working on the foundations of mathematics or in the philosophy of mathematics.

Model Theory
Model Theory

Author: Chen Chung Chang

Publisher: Courier Corporation

Published: 2012-01-01

Total Pages: 674

ISBN-13: 0486488217

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This bestselling textbook for higher-level courses was extensively revised in 1990 to accommodate developments in model theoretic methods. Topics include models constructed from constants, ultraproducts, and saturated and special models. 1990 edition.

Set Theory
Set Theory

Author: Lev D. Beklemishev

Publisher: Elsevier

Published: 2000-04-01

Total Pages: 365

ISBN-13: 0080954863

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Set Theory

Logic from Russell to Church
Logic from Russell to Church

Author: Dov M. Gabbay

Publisher: Elsevier

Published: 2009-06-16

Total Pages: 1069

ISBN-13: 0080885470

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This volume is number five in the 11-volume Handbook of the History of Logic. It covers the first 50 years of the development of mathematical logic in the 20th century, and concentrates on the achievements of the great names of the period--Russell, Post, Gödel, Tarski, Church, and the like. This was the period in which mathematical logic gave mature expression to its four main parts: set theory, model theory, proof theory and recursion theory. Collectively, this work ranks as one of the greatest achievements of our intellectual history. Written by leading researchers in the field, both this volume and the Handbook as a whole are definitive reference tools for senior undergraduates, graduate students and researchers in the history of logic, the history of philosophy, and any discipline, such as mathematics, computer science, and artificial intelligence, for whom the historical background of his or her work is a salient consideration.• The entire range of modal logic is covered• Serves as a singular contribution to the intellectual history of the 20th century• Contains the latest scholarly discoveries and interpretative insights