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This book presents the classic relative consistency proofs in set theory that are obtained by the device of 'inner models'. Three examples of such models are investigated in Chapters VI, VII, and VIII; the most important of these, the class of constructible sets, leads to Gödel's result that the axiom of choice and the continuum hypothesis are consistent with the rest of set theory. The text thus constitutes an introduction to the results of P. Cohen concerning the independence of these axioms, and to many other relative consistency proofs obtained later by Cohen's methods. Chapters I and II introduce the axioms of set theory, and develop such parts of the theory as are indispensable for every relative consistency proof; the method of recursive definition on the ordinals being an important case in point. Although, more or less deliberately, no proofs have been omitted, the development here will be found to require of the reader a certain facility in naive set theory and in the axiomatic method, such e as should be achieved, for example, in first year graduate work (2 cycle de mathernatiques).
The Zermelo/Fraenkel Axioms of Set Theory
Ordinals, Cardinals
The Axiom of Foundation
The Reflection Principle
The Set of Expressions
Ordinal Definable Sets. Relative Consistency of the Axiom of Choice
Fraenkel/Mostowski Models. Relative Consistency of the Negation of the Axiom of Choice (without the Axiom of Foundation)
Constructible Sets. Relative Consistency of the Generalized Continuum Hypothesis