Sy D. Friedman

Fine Structure and Class Forcing - De Gruyter Series in Logic & its Applications Sy D. Friedman Reprint 2011 edition

This book is intended for the student familiar with the basics of axiomatic set theory, including an introduction to Goedel's theory of constructibility. It presents a thorough analysis of the first two approximations to the set-theoretic universe, given by the universes L and L[0#]. Goedel's constructible universe L provides the setting in which the most thorough understanding of set theory can be achieved, through use of the fine structure theory.

The text begins with a streamlined treatment of the fine structure of L, using the notion of S* formula. It follows the technique of forcing with sets or classes, establishing basic facts about the preservation of ZFC and cofinalities. The model L[0#] then arises naturally as a way to select the "relevant" forcing extensions of L. The author shows that forcing, normally a tool for establishing relative consistency results, now becomes a powerful technique for analysing the set-theotic universe. He develops this theme by using class forcing to solve the Genericity, 12-Singleton and Admissibility Spectrum problems of Jensen and Solovay.

The book's further applications of class forcing to genericity, admissibility, descriptive set theory and set-theoretic definability are sure to be of interest to a wide community of set theorists.