Zermelo's Conception of Set Theory
and Reflection Principles
W. W. TAIT
Zermelo ( 1930) is concerned with impredicative second-order set theory. He treats the general case of set theory with urelements, but it will be enough to consider only the case of pure set theory, i.e. without urelements. In this context, Zermelo's theory is the axiomatic second-order theory T2 in the language of pure set theory whose axioms are Extensionality, Regularity, Null Set, Unordered Pairs, Union, Power Set, Second-order Separation, Second-order Replacement, and Choice.1 Thus T2 with the Axiom of Infinity added is the full system T3 of Morse-Kelley (also sometimes called Bernays-Morse) set theory with the Axiom of Choice, which Zermelo regards as a principle of logic. Since the theory is second-order and we are concerned with genuine models of the second-order theory, which Zermelo calls normal domains, and not with arbitrary Henkin models, the power-set operation is absolute and the von Neumann ordinals in a model are well ordered by ∈. So with each model M is associated an ordinal o(M), called the ordinal of M, which is the order type of its set of von Neuman ordinals; and a model M is characterized to within isomorphism by o(M). More generally, the substructures of any two models M and N consisting of the sets____________________
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Publication information: Book title: The Philosophy of Mathematics Today. Contributors: Matthias Schirn - Editor. Publisher: Clarendon Press. Place of publication: Oxford. Publication year: 1998. Page number: 469.