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# Models of Second-Order Zermelo Set Theory

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In [12], Ernst Zermelo described a succession of models for the axioms of set theory as initial segments of a cumulative hierarchy of levels Uα Vα . The recursive definition of the Vα's is:

Thus, a little reflection on the axioms of Zermelo-Fraenkel set theory (ZF) shows that , 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 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 〈Vκ,∈∩(Vκ×Vκ )〉 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 〈Vκ,∈∩(Vκ×Vκ )〉 for κ a strongly inaccessible ordinal.

References
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[1] Bernays, Paul, A system of axiomatic set theory VI, Journal of Symbolic Logic, vol. 13 (1948), pp. 6579.
[2] Bernays, Paul, A system of axiomatic set theory VII, Journal of Symbolic Logic, vol. 19 (1954), pp. 8196.
[3] Boolos, George, The advantages of honest toil over theft, Mathematics and mind, Oxford University Press, 1994.
[4] Drake, Frank, Set theory: An introduction to large cardinals, North-Holland, 1974.
[5] Felgner, Ulrich, Models of ZF-set theory, Lecture Notes in Mathematics, no. 223, Springer-Verlag, Berlin, 1971.
[6] Fraenkel, Abraham, Bar-Hillel, Yehoshua, and Levy, Azriel, Foundations of set theory, North-Holland, 1973.
[7] Levy, Azriel, Basic set theory, Springer-Verlag, 1979.
[8] Montague, Richard, Set theory and higher-order logic, Formal systems and recursive functions (Crossley, J. and Dummett, M., editors), North-Holland, 1967, pp. 131148.
[9] Moschovakis, Yiannis, Notes on set theory, Springer-Verlag, 1994.
[10] Rieger, L., A contribution to Gödel's axiomatic set theory, Czechoslovak Mathematical Journal, vol. 7 (1957), pp. 323357.
[11] Scott, Dana, Axiomatizing set theory, Axiomatic set theory (Jech, Thomas, editor), Proceedings of Symposia in Pure Mathematics, vol. II, American Mathematical Society, 1974, pp. 207214.
[12] Zermelo, Ernst, Über Grenzzahlen und Mengenbereiche: Neue Untersuchungen über die Grundlagen der Mengenlehre, Fundamenta Mathematicae, vol. 16 (1930), pp. 2947.
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Bulletin of Symbolic Logic
• ISSN: 1079-8986
• EISSN: 1943-5894
• URL: /core/journals/bulletin-of-symbolic-logic
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