Combinatorial Set Theory: With a Gentle Introduction to Forcing

ISBN-13: 9783319868127 / Angielski / Miękka / 2019 / 594 str.

Lorenz J. Halbeisen

Opis

This introduction to modern set theory opens the way to advanced current research. Coverage includes the axiom of choice and Ramsey theory, and a detailed explanation of the sophisticated technique of forcing. Offers notes, related results and references.

Recenzje

"Each chapter ends with Notes that often add historical information, offer further remarks on the chapter's contents ... . Halbheisen's Combinatorial Set Theory is an excellent source for the intermediate or advanced student of set theory ... Because of its wealth of material, it should also serve as an excellent resource for those designing advanced courses or searching for seminar assignments for students." (J. M. Plotkin, zbMATH 06755662, 2018)

Spis treści

The Setting.- First-Order Logic in a Nutshell.- Axioms of Set Theory.- Overture: Ramsey's Theorem.- Cardinal Relations in ZF Only.- Forms of Choice.- How to Make Two Balls from One.- Models of Set Theory with Atoms.- Thirteen Cardinals and Their Relations.- The Shattering Number Revisited.- Happy Families and Their Relatives.- Coda: A Dual Form of Ramsey’s Theorem.- The Idea of Forcing.- Martin's Axiom.- The Notion of Forcing.- Proving Unprovability.- Models in Which AC Fails.- Combining Forcing Notions.- Models in Which p=c.- Suslin’s Problem.- Properties of Forcing Extensions.- Cohen Forcing Revisited.- Sacks Forcing.- Silver-Like Forcing Notions.- Miller Forcing.- Mathias Forcing.- How Many Ramsey Ultrafilters Exist?.- Combinatorial Properties of Sets of Partitions.- Suite.

Blurb

This book, now in a thoroughly revised second edition, provides a comprehensive and accessible introduction to modern set theory.

Following an overview of basic notions in combinatorics and first-order logic, the author outlines the main topics of classical set theory in the second part, including Ramsey theory and the axiom of choice. The revised edition contains new permutation models and recent results in set theory without the axiom of choice. The third part explains the sophisticated technique of forcing in great detail, now including a separate chapter on Suslin’s problem. The technique is used to show that certain statements are neither provable nor disprovable from the axioms of set theory. In the final part, some topics of classical set theory are revisited and further developed in light of forcing, with new chapters on Sacks Forcing and Shelah’s astonishing construction of a model with finitely many Ramsey ultrafilters.

Written for graduate students in axiomatic set theory, Combinatorial Set Theory will appeal to all researchers interested in the foundations of mathematics. With extensive reference lists and historical remarks at the end of each chapter, this book is suitable for self-study.