The stationary tower: notes on a course by W. Hugh Woodin

The stationary tower is an important method in modern set theory, invented by Hugh Woodin in the 1980s. It is a means of constructing generic elementary embeddings and can be applied to produce a variety of useful forcing effects. Hugh Woodin is a leading figure in modern set theory, having made many deep and lasting contributions to the field, in particular to descriptive set theory and large cardinals. This book is the first detailed treatment of his method of the stationary tower that is generally accessible to graduate students in mathematical logic. By giving complete proofs of all the main theorems and discussing them in context, it is intended that the book will become the standard reference on the stationary tower and its applications to descriptive set theory. The first two chapters are taken from a graduate course Woodin taught at Berkeley. The concluding theorem in the course was that large cardinals imply that all sets of reals in the smallest model of set theory (without choice) containing the reals are Lebesgue measurable. Additional sections include a proof (using the stationary tower) of Woodin's theorem that, with large cardinals, the Continuum Hypothesis settles all questions of the same complexity as well as some of Woodin's applications of the stationary tower to the studies of absoluteness and determinacy. The book is suitable for a graduate course that assumes some familiarity with forcing, constructibility, and ultrapowers. It is also recommended for researchers interested in logic, set theory, and forcing.