The Ultrapower Axiom

The Ultrapower Axiom
Author :
Publisher : Walter de Gruyter GmbH & Co KG
Total Pages : 424
Release :
ISBN-10 : 9783110719796
ISBN-13 : 3110719797
Rating : 4/5 (96 Downloads)

Synopsis The Ultrapower Axiom by : Gabriel Goldberg

The book is about strong axioms of infi nity in set theory (also known as large cardinal axioms), and the ongoing search for natural models of these axioms. Assuming the Ultrapower Axiom, a combinatorial principle conjectured to hold in all such natural models, we solve various classical problems in set theory (for example, the Generalized Continuum Hypothesis) and uncover a theory of large cardinals that is much clearer than the one that can be developed using only the standard axioms.

Trends in Set Theory

Trends in Set Theory
Author :
Publisher : American Mathematical Soc.
Total Pages : 207
Release :
ISBN-10 : 9781470443320
ISBN-13 : 1470443325
Rating : 4/5 (20 Downloads)

Synopsis Trends in Set Theory by : Samuel Coskey

This volume contains the proceedings of Simon Fest, held in honor of Simon Thomas's 60th birthday, from September 15–17, 2017, at Rutgers University, Piscataway, New Jersey. The topics covered showcase recent advances from a variety of main areas of set theory, including descriptive set theory, forcing, and inner model theory, in addition to several applications of set theory, including ergodic theory, combinatorics, and model theory.

The Philosophy of Penelope Maddy

The Philosophy of Penelope Maddy
Author :
Publisher : Springer Nature
Total Pages : 526
Release :
ISBN-10 : 9783031584251
ISBN-13 : 3031584252
Rating : 4/5 (51 Downloads)

Synopsis The Philosophy of Penelope Maddy by : Sophia Arbeiter

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

The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal
Author :
Publisher : Walter de Gruyter
Total Pages : 944
Release :
ISBN-10 : 9783110804737
ISBN-13 : 3110804735
Rating : 4/5 (37 Downloads)

Synopsis The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal by : W. Hugh Woodin

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
Author :
Publisher : Walter de Gruyter GmbH & Co KG
Total Pages : 325
Release :
ISBN-10 : 9783110719734
ISBN-13 : 3110719738
Rating : 4/5 (34 Downloads)

Synopsis The Ultrapower Axiom by : Gabriel Goldberg

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.

Extensions of the Axiom of Determinacy

Extensions of the Axiom of Determinacy
Author :
Publisher : American Mathematical Society
Total Pages : 182
Release :
ISBN-10 : 9781470472108
ISBN-13 : 1470472104
Rating : 4/5 (08 Downloads)

Synopsis Extensions of the Axiom of Determinacy by : Paul B. Larson

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.

The Determinacy of Long Games

The Determinacy of Long Games
Author :
Publisher : Walter de Gruyter
Total Pages : 333
Release :
ISBN-10 : 9783110200065
ISBN-13 : 3110200066
Rating : 4/5 (65 Downloads)

Synopsis The Determinacy of Long Games by : Itay Neeman

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.

New Infinitary Mathematics

New Infinitary Mathematics
Author :
Publisher : Charles University in Prague, Karolinum Press
Total Pages : 352
Release :
ISBN-10 : 9788024646633
ISBN-13 : 8024646633
Rating : 4/5 (33 Downloads)

Synopsis New Infinitary Mathematics by : Petr Vopěnka

The dominant current of twentieth-century mathematics, which simultaneously explores and applies infinity (albeit in bizarre ideal worlds), relies on Cantor's classical theory of infinite sets. Cantor’s theory in turn relies on the problematic assumption of the existence of the set of all natural numbers, the only justification for which – a theological justification - is usually concealed and pushed into the collective unconscious. This book begins by surveying the theological background, emergence, and development of classical set theory. The author warns us about the dangers implicit in the construction of set theory, traceable in his own and other eminent mathematicians' seminal works on the subject. He then goes on to present an argument about the absurdity of the assumption of the existence of the set of all natural numbers. However, the author’s contribution is not just a negation of current views and assumptions. On the contrary, the new infinitary mathematics that he proceeds to propose and develop is driven by a cautious effort to transcend the horizon bounding the ancient geometric world and pre-set-theoretical mathematics, whilst allowing mathematics to correspond more closely to the natural real world surrounding us. The final parts are devoted to a discussion of real numbers and to demonstrating how, within the new infinitary mathematics, calculus can be rehabilitated in its original form employing infinitesimals.

Ernst Zermelo - Collected Works/Gesammelte Werke

Ernst Zermelo - Collected Works/Gesammelte Werke
Author :
Publisher : Springer Science & Business Media
Total Pages : 673
Release :
ISBN-10 : 9783540793847
ISBN-13 : 3540793844
Rating : 4/5 (47 Downloads)

Synopsis Ernst Zermelo - Collected Works/Gesammelte Werke by : Ernst Zermelo

Ernst Zermelo (1871-1953) is regarded as the founder of axiomatic set theory and best-known for the first formulation of the axiom of choice. However, his papers include also pioneering work in applied mathematics and mathematical physics. This edition of his collected papers will consist of two volumes. Besides providing a biography, the present Volume I covers set theory, the foundations of mathematics, and pure mathematics and is supplemented by selected items from his Nachlass and part of his translations of Homer's Odyssey. Volume II will contain his work in the calculus of variations, applied mathematics, and physics. The papers are each presented in their original language together with an English translation, the versions facing each other on opposite pages. Each paper or coherent group of papers is preceded by an introductory note provided by an acknowledged expert in the field which comments on the historical background, motivations, accomplishments, and influence.

Automatic Complexity

Automatic Complexity
Author :
Publisher : Walter de Gruyter GmbH & Co KG
Total Pages : 156
Release :
ISBN-10 : 9783110774870
ISBN-13 : 3110774879
Rating : 4/5 (70 Downloads)

Synopsis Automatic Complexity by : Bjørn Kjos-Hanssen