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- W1125945453 abstract "This chapter reviews the development of Zermelo–Fraenkel set theory, ZF. The theory begins with a collection M of objects called urelements. The theory constructs on the collection M in stages. At each stage α, all urelements and all sets constructed at previous stages are available. A collection is a set if it is formed at some stage in this construction; the collection of all sets built on M is denoted by V M . The chapter introduces the axioms of KPU. The theory KPU is formulated relative to a language L* = L(∈, …) which extends L by adding a membership symbol ∈ and, possibly, other function, relation and constant symbols. The axioms of KPU are of three kinds. The axioms of extensionality and foundation concern the basic nature of sets. A set a is transitive if for all x ∈ a and all y ∈ x, y ∈ a. Thus if a is a set of urelements it is transitive. An urelement is never transitive because only sets are transitive. It can be proved in KPU that for every x there is a unique transitive set a with x ⊆ a such that if b is any other transitive set containing x , then a ⊆ b." @default.
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- W1125945453 date "1974-01-01" @default.
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- W1125945453 title "Admissible Sets Over Models of Set Theory" @default.
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- W1125945453 doi "https://doi.org/10.1016/s0049-237x(08)70584-6" @default.
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