Combinatorial Set Theory : With a Gentle Introduction to Forcing / by Lorenz J. Halbeisen.

Format:

Book

Author/Creator:

Halbeisen, Lorenz J., Author.

Series:

Springer Monographs in Mathematics, 2196-9922

Language:

English

Subjects (All):

Logic, Symbolic and mathematical.

Discrete mathematics.

Mathematical Logic and Foundations.

Discrete Mathematics.

Local Subjects:

Mathematical Logic and Foundations.

Discrete Mathematics.

Physical Description:

1 online resource (XVI, 594 p. 20 illus.)

Edition:

2nd ed. 2017.

Place of Publication:

Cham : Springer International Publishing : Imprint: Springer, 2017.

Summary:

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. Writtenfor 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.

Contents:

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.

Notes:

Includes bibliographical references at the end of each chapters and indexes.

ISBN:

3-319-60231-4