Inner Models and Large Cardinals

Inner Models and Large Cardinals
Author :
Publisher : Walter de Gruyter
Total Pages : 385
Release :
ISBN-10 : 9783110857818
ISBN-13 : 3110857812
Rating : 4/5 (18 Downloads)

Synopsis Inner Models and Large Cardinals by : Martin Zeman

This volume is an introduction to inner model theory, an area of set theory which is concerned with fine structural inner models reflecting large cardinal properties of the set theoretic universe. The monograph contains a detailed presentation of general fine structure theory as well as a modern approach to the construction of small core models, namely those models containing at most one strong cardinal, together with some of their applications. The final part of the book is devoted to a new approach encompassing large inner models which admit many Woodin cardinals. The exposition is self-contained and does not assume any special prerequisities, which should make the text comprehensible not only to specialists but also to advanced students in Mathematical Logic and Set Theory.

Inner Models and Large Cardinals

Inner Models and Large Cardinals
Author :
Publisher : Walter de Gruyter
Total Pages : 392
Release :
ISBN-10 : 3110163683
ISBN-13 : 9783110163681
Rating : 4/5 (83 Downloads)

Synopsis Inner Models and Large Cardinals by : Martin Zeman

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 Higher Infinite

The Higher Infinite
Author :
Publisher : Springer Science & Business Media
Total Pages : 555
Release :
ISBN-10 : 9783540888673
ISBN-13 : 3540888675
Rating : 4/5 (73 Downloads)

Synopsis The Higher Infinite by : Akihiro Kanamori

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.

Set Theory

Set Theory
Author :
Publisher : Elsevier
Total Pages : 365
Release :
ISBN-10 : 9780080954868
ISBN-13 : 0080954863
Rating : 4/5 (68 Downloads)

Synopsis Set Theory by : Lev D. Beklemishev

Set Theory

Set Theory

Set Theory
Author :
Publisher : Springer
Total Pages : 335
Release :
ISBN-10 : 9783319067254
ISBN-13 : 3319067257
Rating : 4/5 (54 Downloads)

Synopsis Set Theory by : Ralf Schindler

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.

Handbook of Set Theory

Handbook of Set Theory
Author :
Publisher : Springer Science & Business Media
Total Pages : 2200
Release :
ISBN-10 : 9781402057649
ISBN-13 : 1402057644
Rating : 4/5 (49 Downloads)

Synopsis Handbook of Set Theory by : Matthew Foreman

Numbers imitate space, which is of such a di?erent nature —Blaise Pascal It is fair to date the study of the foundation of mathematics back to the ancient Greeks. The urge to understand and systematize the mathematics of the time led Euclid to postulate axioms in an early attempt to put geometry on a ?rm footing. With roots in the Elements, the distinctive methodology of mathematics has become proof. Inevitably two questions arise: What are proofs? and What assumptions are proofs based on? The ?rst question, traditionally an internal question of the ?eld of logic, was also wrestled with in antiquity. Aristotle gave his famous syllogistic s- tems, and the Stoics had a nascent propositional logic. This study continued with ?ts and starts, through Boethius, the Arabs and the medieval logicians in Paris and London. The early germs of logic emerged in the context of philosophy and theology. The development of analytic geometry, as exempli?ed by Descartes, ill- tratedoneofthedi?cultiesinherentinfoundingmathematics. Itisclassically phrased as the question ofhow one reconciles the arithmetic with the geom- ric. Arenumbers onetypeofthingand geometricobjectsanother? Whatare the relationships between these two types of objects? How can they interact? Discovery of new types of mathematical objects, such as imaginary numbers and, much later, formal objects such as free groups and formal power series make the problem of ?nding a common playing ?eld for all of mathematics importunate. Several pressures made foundational issues urgent in the 19th century.

Naturalism in Mathematics

Naturalism in Mathematics
Author :
Publisher : Clarendon Press
Total Pages : 265
Release :
ISBN-10 : 9780191518973
ISBN-13 : 0191518972
Rating : 4/5 (73 Downloads)

Synopsis Naturalism in Mathematics by : Penelope Maddy

Our much-valued mathematical knowledge rests on two supports: the logic of proof and the axioms from which those proofs begin. Naturalism in Mathematics investigates the status of the latter, the fundamental assumptions of mathematics. These were once held to be self-evident, but progress in work on the foundations of mathematics, especially in set theory, has rendered that comforting notion obsolete. Given that candidates for axiomatic status cannot be proved, what sorts of considerations can be offered for or against them? That is the central question addressed in this book. One answer is that mathematics aims to describe an objective world of mathematical objects, and that axiom candidates should be judged by their truth or falsity in that world. This promising view—realism—is assessed and finally rejected in favour of another—naturalism—which attends less to metaphysical considerations of objective truth and falsity, and more to practical considerations drawn from within mathematics itself. Penelope Maddy defines this naturalism, explains the motivation for it, and shows how it can be helpfully applied in the assessment of candidates for axiomatic status in set theory. Maddy's clear, original treatment of this fundamental issue is informed by current work in both philosophy and mathematics, and will be accessible and enlightening to readers from both disciplines.

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.

The Core Model Iterability Problem

The Core Model Iterability Problem
Author :
Publisher : Cambridge University Press
Total Pages : 119
Release :
ISBN-10 : 9781107167964
ISBN-13 : 1107167965
Rating : 4/5 (64 Downloads)

Synopsis The Core Model Iterability Problem by : John R. Steel

Suitable for graduate students and researchers in set theory, this volume develops a method for constructing core models that have Woodin cardinals.

Higher Set Theory

Higher Set Theory
Author :
Publisher : Springer
Total Pages : 481
Release :
ISBN-10 : 9783540357490
ISBN-13 : 3540357491
Rating : 4/5 (90 Downloads)

Synopsis Higher Set Theory by : G.H. Müller