The Forcing Method in Set Theory: An Introduction via Boolean Valued Logic

The main aim of this book is to provide a compact self-contained presentation of the forcing technique devised by Cohen to establish the independence of the continuum hypothesis from the axioms of set theory. The book follows the approach to the forcing technique via Boolean valued semantics independently introduced by Vopenka and Scott/Solovay; it develops out of notes I prepared for several master courses on this and related topics and aims to provide an alternative (and more compact) account of this topic with respect to the available classical textbooks. The aim of the book is to take up a reader with familiarity with logic and set theory at the level of an undergraduate course on both topics (e.g., familiar with most of the content of introductory books on first-order logic and set theory) and bring her/him to page with the use of the forcing method to produce independence (or undecidability results) in mathematics. Familiarity of the reader with general topology would also be quite helpful; however, the book provides a compact account of all the needed results on this matter. Furthermore, the book is organized in such a way that many of its parts can also be read by scholars with almost no familiarity with first-order logic and/or set theory. The book presents the forcing method outlining, in many situations, the intersections of set theory and logic with other mathematical domains. My hope is that this book can be appreciated by scholars in set theory and by readers with a mindset oriented towards areas of mathematics other than logic and a keen interest in the foundations of mathematics.

Matteo Viale is a full professor in mathematical logic in the Mathematics Department of the University of Torino. In 2006, he won the Sacks prize in mathematical logic awarded by the Association of Symbolic Logic for the best PhD thesis in logic for that year. He has also won the 2010 Kurt Goedel Research fellowship (awarded by the Kurt Goedel Society) and the Fubini prize in 2011 (awarded by the Istituto Guido Boella). He has published over 20 papers in refereed journals, including top ones such as JAMS, TAMS, and Advances in Mathematics. He has taught master-level courses in set theory since 2012. He has supervised over 20 master’s theses in mathematical logic and three PhD students.