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Models of Set Theory

DOI: 10.1016/s0049-237x(09)70459-8
  • Logic
  • Mathematics


Publisher Summary This chapter examines the aim of the various lectures, which is to outline various methods used in construction of models for axioms of set theory. In the introductory lecture I, the chapter describes three systems of axioms for abstract set theory. In all these systems there are two primitive notions: class and membership. Lecture II discusses the stratification of the universe into levels. In lecture III, the Scott–Scarpellini theorem is applied to obtain various families of sets that form models of Zermelo–Fraenkel (ZF). Lecture IV is devoted to models of ZF. It introduces the notion of height and width of a model and compares various models as to their height and width. Lecture V introduces a new kind of models to establish the independence of various set-theoretical hypotheses from the axioms of ZF Scott. Lecture VI proves two theorems: all formulae provable in ZF are valid in an arbitrary Boolean model, and the axiom of choice is valid in an arbitrary Boolean model. Lecture VII constructs a model in which the axiom of constructability is not valid. Lecture VIII constructs a model in which the continuum hypothesis is false.

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