This book introduces the theory of inner models of set theory constructible relative to coherent extender sequences: such models admit a fine structure analogous to that of Gödel’s constructible universe. Chapters 1-3 introduce the general fine structure theory of acceptable structures, developed abstractly without reference to large cardinals or inner models. Chapters 4-7 present full core model theory for measures of order 0, and Chapter 8 indicates a generalisation for models that can contain up to one strong cardinal. A ‘linear’ iterability (a possibility of forming certain required iterated ultrapowers), associated to the models in question, plays a crucial role here. As Chapter 9 explains, this iterability is highly non-linear in the case of theory of Jensen extender models that are beyond one strong cardinal.

The exposition is relatively self-contained, but some elementary knowledge (for example, concerning acceptable structures) is assumed. The book can serve as an introductory text for those unfamiliar with the subject and as a textbook for graduate students.

Reviewer:

jmlc