The Philosophy of Mathematics Today

By Matthias Schirn | Go to book overview

18
Zermelo's Conception of Set Theory
and Reflection Principles

W. W. TAIT


1. ZERMELO 1930

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

____________________
This is a modified version of the talk that I gave at the conference 'Philosophy of Mathematics Today'.
1
Second-order Separation is a consequence of the other axioms, but it will be useful to retain it as an axiom. There is some question about what Zermelo intended. In fn. 1 of ( 1930), Zermelo asserts that the Axioms of Separation, yielding {xt | F(x)}, and Replacement, yielding {G(x) | xt}, for any set t, are to hold for arbitrary propositional function F and function G. But then he goes on to refer to the debate over the notion of 'definiteness' and to Zermelo ( 1929) in which he characterizes the definite functions in effect as those definable by second- order formulas of set theory. If he intended his arbitrary F and G to be restricted in this way to those which are definite, then there are Henkin models of T2 (as John Burgess pointed out to me), but not genuine models. In this case, his arguments for the results summarized below are invalid. So I will assume that he was referring to arbitrary F and G in genuine models and that the reference to Zermelo ( 1929) was simply a mistake. For example, how could an isomorphism from a model M to a model N with o(M) = o(N) be defined when Replacement in M is restricted to functions that are second-order definable on M?

-469-

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