Combinatorial Set Theory: With a Gentle Introduction to Forcing

ISBN-13: 9781447121725 / Angielski / Twarda / 2011 / 456 str.

Lorenz Halbeisen

Opis

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.

Recenzje

From the reviews:

"The text is very well organised; each chapter ends with useful notes containing much historical background and the source of the main results, together with reports on related results and the references. The beautifully written book is intended for students on graduate courses in axiomatic set theory, and it is also excellent as a text for self-study." (Peter Shiu, The Mathematical Gazette, Vol. 98 (541), March, 2014)

"The book under review provides a thorough and nicely written account of combinatorial set theory and infinite Ramsey theory together with a treatment of the underlying set theoretical axioms as well as of sophisticated methods which are involved in proving independence results. ... I can recommend this book to all graduate students, PostDocs, and researchers who are interested in set theoretical combinatorics ... . also mathematicians from other areas who are interested in the foundational aspects of their subject will enjoy this book." (Ralf Schindler, Jahresbericht der Deutschen Mathematiker-Vereinigung, Vol. 115, 2013)

"This book provides a self-contained introduction to axiomatic set theory with main focus on infinitary combinatorics and the forcing technique. It is intended as a textbook in courses as well as for self-study. ... The author gives the historical background and the sources of the main results in the Notes of each chapter. He also gives hints for further studies in his sections 'Related results'." (Martin Weese, Zentralblatt MATH, Vol. 1237, 2012)

Spis treści

The Setting.- Overture: Ramsey's Theorem.- The Axioms of Zermelo-Fraenkel Set Theory.- Cardinal Relations in ZF only.- The Axiom of Choice.- How to Make Two Balls from One.- Models of Set Theory with Atoms.- Twelve 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.- Models of Finite Fragments of Set Theory.- Proving Unprovability.- Models in which AC Fails.- Combining Forcing Notions.- Models in which p = c.- Properties of Forcing Extensions.- Cohen Forcing Revisited.- Silver-Like Forcing Notions.- Miller Forcing.- Mathias Forcing.- On the Existence of Ramsey Ultrafilters.- Combinatorial Properties of Sets of Partitions.- Suite.