Combinatorial Set Theory: With a Gentle Introduction to Forcing

✍ Scribed by Lorenz J. Halbeisen (auth.)

• Publisher: Springer-Verlag London
• Year: 2012
• Tongue: English
• Leaves: 448
• Series: Springer Monographs in Mathematics
• Edition: 1
• Category: Library

✦ Synopsis

This book provides a self-contained introduction to modern set theory and also opens up some more advanced areas of current research in this field. The first part offers an overview of classical set theory wherein the focus lies on the axiom of choice and Ramsey theory. In the second part, the sophisticated technique of forcing, originally developed by Paul Cohen, is explained in great detail. With this technique, one can show that certain statements, like the continuum hypothesis, are neither provable nor disprovable from the axioms of set theory. In the last part, some topics of classical set theory are revisited and further developed in the light of forcing. The notes at the end of each chapter put the results in a historical context, and the numerous related results and the extensive list of references lead the reader to the frontier of research. This book will appeal to all mathematicians interested in the foundations of mathematics, but will be of particular use to graduates in this field.

✦ Table of Contents

Front Matter....Pages I-XVI
Front Matter....Pages 7-7
The Setting....Pages 1-6
Front Matter....Pages 7-7
Overture: Ramsey’s Theorem....Pages 9-24
The Axioms of Zermelo–Fraenkel Set Theory....Pages 25-70
Cardinal Relations in ZF Only....Pages 71-100
The Axiom of Choice....Pages 101-141
How to Make Two Balls from One....Pages 143-155
Models of Set Theory with Atoms....Pages 157-177
Twelve Cardinals and Their Relations....Pages 179-199
The Shattering Number Revisited....Pages 201-213
Happy Families and Their Relatives....Pages 215-233
Coda: A Dual Form of Ramsey’s Theorem....Pages 235-255
Front Matter....Pages 257-257
The Idea of Forcing....Pages 259-261
Martin’s Axiom....Pages 263-272
The Notion of Forcing....Pages 273-293
Models of Finite Fragments of Set Theory....Pages 295-303
Proving Unprovability....Pages 305-310
Models in Which AC Fails....Pages 311-326
Combining Forcing Notions....Pages 327-345
Models in Which $\mathfrak {p}=\mathfrak {c}$ ....Pages 347-354
Front Matter....Pages 355-355
Properties of Forcing Extensions....Pages 357-364
Front Matter....Pages 355-355
Cohen Forcing Revisited....Pages 365-375
Silver-Like Forcing Notions....Pages 377-382
Miller Forcing....Pages 383-394
Mathias Forcing....Pages 395-404
On the Existence of Ramsey Ultrafilters....Pages 405-417
Combinatorial Properties of Sets of Partitions....Pages 419-430
Suite....Pages 431-437
Back Matter....Pages 439-453

✦ Subjects

Mathematical Logic and Foundations