Artigo Revisado por pares

Models of Second-Order Zermelo Set Theory

1999; Cambridge University Press; Volume: 5; Issue: 3 Linguagem: Inglês

10.2307/421182

ISSN

1943-5894

Autores

Gabriel Uzquiano,

Tópico(s)

Advanced Algebra and Logic

Resumo

In [12], Ernst Zermelo described a succession of models for the axioms of set theory as initial segments of a cumulative hierarchy of levels U α V α . The recursive definition of the V α ' s is: Thus, a little reflection on the axioms of Zermelo-Fraenkel set theory ( ZF ) shows that Vω , the first transfinite level of the hierarchy, is a model of all the axioms of ZF with the exception of the axiom of infinity. And, in general, one finds that if κ is a strongly inaccessible ordinal, then Vκ is a model of all of the axioms of ZF . (For all these models, we take ∈ to be the standard element-set relation restricted to the members of the domain.) Doubtless, when cast as a first-order theory, ZF does not characterize the structures 〈 V κ ,∈∩ ( V κ ×V κ )〉 for κ a strongly inaccessible ordinal, by the Löwenheim-Skolem theorem. Still, one of the main achievements of [12] consisted in establishing that a characterization of these models can be attained when one ventures into second-order logic. For let second-order ZF be, as usual, the theory that results from ZF when the axiom schema of replacement is replaced by its second-order universal closure. Then, it is a remarkable result due to Zermelo that second-order ZF can only be satisfied in models of the form 〈 V κ ,∈∩ ( V κ ×V κ )〉 for κ a strongly inaccessible ordinal.

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