Fine structure and class forcing
The major theme of this monograph is that class forcing is not just a tool for proving consistency results, it is a way to prove absolute results in the theory \(\text{ZFC}+0^{\sharp}\) exists. The book provides a detailed analysis of the inner models \(L\) and \(L[0^{\sharp}]\). The reader is assumed to be familiar with the basics of ZFC and the constructible universe \(L\); however, no background in either forcing or fine structure is presumed. The book's opening chapter, the \(\Sigma^{*}\) approach to the fine structure of \(L\), is a reformulation of Jensen's fine structure theory. It ends with proofs of the \(\square\) and Fine Scale Principles and the construction of morasses. Chapters 2, 3, and 4 are the heart of the book. They give the basic theory of class forcing and they include discussions of generic existence and Jensen coding. Chapters 5, 6, 7, and 8 are essentially independent of each other. They present applications of class forcing to settle problems about genericity, admissibility, descriptive set theory and set-theoretic definability. The titles of these latter chapters are: The Genericity Problem; The \(\Pi_{2}^{1}\) Singleton Problem; The Admissibility Spectrum Problem; Further Applications of Class Forcing. The monograph ends with a list of 16 open problems. Reviewer's remark: In spite of the technical nature of its material, this monograph seems highly readable.
scientific article; zbMATH DE number 1493050