The Determinacy of Long Games

The Determinacy of Long Games PDF

Author: Itay Neeman

Publisher: Walter de Gruyter

Published: 2008-08-22

Total Pages: 333

ISBN-13: 3110200066

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In this volume the author develops and applies methods for proving, from large cardinals, the determinacy of definable games of countable length on natural numbers. The determinacy is ultimately derived from iteration strategies, connecting games on natural numbers with the specific iteration games that come up in the study of large cardinals. The games considered in this text range in strength, from games of fixed countable length, through games where the length is clocked by natural numbers, to games in which a run is complete when its length is uncountable in an inner model (or a pointclass) relative to the run. More can be done using the methods developed here, reaching determinacy for games of certain length. The book is largely self-contained. Only graduate level knowledge of modern techniques in large cardinals and basic forcing is assumed. Several exercises allow the reader to build on the results in the text, for example connecting them with universally Baire and homogeneously Suslin sets. - Important contribution to one of the main features of current set theory, as initiated and developed by Jensen, Woodin, Steel and others.

Logic Colloquium '01

Logic Colloquium '01 PDF

Author: Matthias Baaz

Publisher: Cambridge University Press

Published: 2017-03-30

Total Pages:

ISBN-13: 1108695442

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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. This volume, the twentieth publication in the Lecture Notes in Logic series, contains the proceedings of the 2001 European Summer Meeting of the Association for Symbolic Logic, held at the Vienna University of Technology. Two long articles present accessible expositions on resolution theorem proving and the determinacy of long games. The remaining articles cover separate research topics in many areas of mathematical logic, including applications in computer science, proof theory, set theory, model theory, computability theory, linguistics and aspects of philosophy. This collection will interest not only mathematical logicians but also philosophical logicians, historians of logic, computer scientists, formal linguists and mathematicians working in algebra, abstract analysis and topology.

Sets and Proofs

Sets and Proofs PDF

Author: S. Barry Cooper

Publisher: Cambridge University Press

Published: 1999-06-17

Total Pages: 450

ISBN-13: 9780521635493

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First of two volumes providing a comprehensive guide to mathematical logic.

The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal

The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal PDF

Author: W. Hugh Woodin

Publisher: Walter de Gruyter

Published: 2013-02-01

Total Pages: 944

ISBN-13: 3110804735

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The series is devoted to the publication of high-level monographs on all areas of mathematical logic and its applications. It is addressed to advanced students and research mathematicians, and may also serve as a guide for lectures and for seminars at the graduate level.

The Ultrapower Axiom

The Ultrapower Axiom PDF

Author: Gabriel Goldberg

Publisher: Walter de Gruyter GmbH & Co KG

Published: 2022-03-21

Total Pages: 325

ISBN-13: 3110719738

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The book is about strong axioms of infinity (also known as large cardinal axioms) in set theory, and the ongoing search for natural models of these axioms. Assuming the Ultrapower Axiom, we solve various classical problems in set theory (e.g., the Generalized Continuum Hypothesis) and develop a theory of large cardinals that is much clearer than the theory that can be developed using only the standard axioms.

Set Theory

Set Theory PDF

Author: Ralf Schindler

Publisher: Springer

Published: 2014-05-22

Total Pages: 335

ISBN-13: 3319067257

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This textbook gives an introduction to axiomatic set theory and examines the prominent questions that are relevant in current research in a manner that is accessible to students. Its main theme is the interplay of large cardinals, inner models, forcing and descriptive set theory. The following topics are covered: • Forcing and constructability • The Solovay-Shelah Theorem i.e. the equiconsistency of ‘every set of reals is Lebesgue measurable’ with one inaccessible cardinal • Fine structure theory and a modern approach to sharps • Jensen’s Covering Lemma • The equivalence of analytic determinacy with sharps • The theory of extenders and iteration trees • A proof of projective determinacy from Woodin cardinals. Set Theory requires only a basic knowledge of mathematical logic and will be suitable for advanced students and researchers.

Interpreting Gödel

Interpreting Gödel PDF

Author: Juliette Kennedy

Publisher: Cambridge University Press

Published: 2014-08-21

Total Pages: 293

ISBN-13: 1139991752

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The logician Kurt Gödel (1906–1978) published a paper in 1931 formulating what have come to be known as his 'incompleteness theorems', which prove, among other things, that within any formal system with resources sufficient to code arithmetic, questions exist which are neither provable nor disprovable on the basis of the axioms which define the system. These are among the most celebrated results in logic today. In this volume, leading philosophers and mathematicians assess important aspects of Gödel's work on the foundations and philosophy of mathematics. Their essays explore almost every aspect of Godel's intellectual legacy including his concepts of intuition and analyticity, the Completeness Theorem, the set-theoretic multiverse, and the state of mathematical logic today. This groundbreaking volume will be invaluable to students, historians, logicians and philosophers of mathematics who wish to understand the current thinking on these issues.

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