The stationary tower. Notes on a course by W. Hugh Woodin. (English) Zbl 1072.03031
University Lecture Series 32. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3604-8/pbk). x, 132 p. (2004).
The stationary tower was invented by Woodin in the 1980s. It is an important tool in modern set theory and is used in the construction of generic elementary embeddings. It allows one to produce useful forcing conditions and has many deep applications in descriptive set theory. This book is an introduction to stationary tower forcing. It is mostly based on lecture notes given by Woodin in the Spring of 1996.
In Chapter 1 the author presents some relevant background material. He investigates elementary embeddings. He starts with ultrapowers and with trees of measures and goes on with extenders, Woodin cardinals, generic ultrapowers and the nonstationary ideal.
In Chapter 2 the author investigates the stationary tower in detail and develops its basic properties. He examines completely Jonsson cardinals and gives some forcing applications. Then he goes on with well-foundedness, preserving Wooding cardinals and the countable tower.
In Chapter 3 he presents applications. So he investigates regularity properties and absoluteness, tree representations, and fixing the theory of the universally Baire sets.
This book can be used as a reference on the stationary tower and its applications to descriptive set theory.
In Chapter 1 the author presents some relevant background material. He investigates elementary embeddings. He starts with ultrapowers and with trees of measures and goes on with extenders, Woodin cardinals, generic ultrapowers and the nonstationary ideal.
In Chapter 2 the author investigates the stationary tower in detail and develops its basic properties. He examines completely Jonsson cardinals and gives some forcing applications. Then he goes on with well-foundedness, preserving Wooding cardinals and the countable tower.
In Chapter 3 he presents applications. So he investigates regularity properties and absoluteness, tree representations, and fixing the theory of the universally Baire sets.
This book can be used as a reference on the stationary tower and its applications to descriptive set theory.
Reviewer: Martin Weese (Potsdam)
MathOverflow Questions:
Consequences of existence of a certain function from \(\omega_1\) to \(\omega_1\)MSC:
| 03E40 | Other aspects of forcing and Boolean-valued models |
| 03E15 | Descriptive set theory |
| 03E35 | Consistency and independence results |
| 03E55 | Large cardinals |
| 03E60 | Determinacy principles |
| 03-02 | Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations |
