The Higher Infinite

The Higher Infinite PDF

Author: Akihiro Kanamori

Publisher: Springer Science & Business Media

Published: 2008-11-23

Total Pages: 555

ISBN-13: 3540888675

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Over the years, this book has become a standard reference and guide in the set theory community. It provides a comprehensive account of the theory of large cardinals from its beginnings and some of the direct outgrowths leading to the frontiers of contemporary research, with open questions and speculations throughout.

Zermelo's Axiom of Choice

Zermelo's Axiom of Choice PDF

Author: Gregory H. Moore

Publisher: Courier Corporation

Published: 2012-09-20

Total Pages: 450

ISBN-13: 0486488411

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"This book chronicles the work of mathematician Ernst Zermelo (1871-1953) and his development of set theory's crucial principle, the axiom of choice. It covers the axiom's formulation during the early 20th century, the controversy it engendered, and its current central place in set theory and mathematical logic. 1982 edition"--

Axiom of Choice

Axiom of Choice PDF

Author: Horst Herrlich

Publisher: Springer

Published: 2006-07-21

Total Pages: 207

ISBN-13: 3540342680

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AC, the axiom of choice, because of its non-constructive character, is the most controversial mathematical axiom. It is shunned by some, used indiscriminately by others. This treatise shows paradigmatically that disasters happen without AC and they happen with AC. Illuminating examples are drawn from diverse areas of mathematics, particularly from general topology, but also from algebra, order theory, elementary analysis, measure theory, game theory, and graph theory.

Quine, New Foundations, and the Philosophy of Set Theory

Quine, New Foundations, and the Philosophy of Set Theory PDF

Author: Sean Morris

Publisher: Cambridge University Press

Published: 2018-12-13

Total Pages: 221

ISBN-13: 1108604536

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Quine's set theory, New Foundations, has often been treated as an anomaly in the history and philosophy of set theory. In this book, Sean Morris shows that it is in fact well-motivated, emerging in a natural way from the early development of set theory. Morris introduces and explores the notion of set theory as explication: the view that there is no single correct axiomatization of set theory, but rather that the various axiomatizations all serve to explicate the notion of set and are judged largely according to pragmatic criteria. Morris also brings out the important interplay between New Foundations, Quine's philosophy of set theory, and his philosophy more generally. We see that his early technical work in logic foreshadows his later famed naturalism, with his philosophy of set theory playing a crucial role in his primary philosophical project of clarifying our conceptual scheme and specifically its logical and mathematical components.

The Principles of Mathematics Revisited

The Principles of Mathematics Revisited PDF

Author: Jaakko Hintikka

Publisher: Cambridge University Press

Published: 1998-04-28

Total Pages: 308

ISBN-13: 9780521624985

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This book, written by one of philosophy's pre-eminent logicians, argues that many of the basic assumptions common to logic, philosophy of mathematics and metaphysics are in need of change. It is therefore a book of critical importance to logical theory. Jaakko Hintikka proposes a new basic first-order logic and uses it to explore the foundations of mathematics. This new logic enables logicians to express on the first-order level such concepts as equicardinality, infinity, and truth in the same language. The famous impossibility results by Gödel and Tarski that have dominated the field for the last sixty years turn out to be much less significant than has been thought. All of ordinary mathematics can in principle be done on this first-order level, thus dispensing with the existence of sets and other higher-order entities.

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