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Constructible models of subsystems of ZF

  • Richard Gostanian (a1)
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  • Published online: 01 March 2014

One of the main results of Gödel [4] and [5] is that, if M is a transitive set such that 〈M, Є 〉 is a model of ZF (Zermelo-Fraenkel set theory) and α is the least ordinal not in M, then 〈Lα, Є 〉 is also a model of ZF.

In this note we shall use the Jensen uniformisation theorem to show that results analogous to the above hold for certain subsystems of ZF. The subsystems we have in mind are those that are formed by restricting the formulas in the separation and replacement axioms to various levels of the Levy hierarchy.

This is all done in §1. In §2 we proceed to establish the exact order relationships which hold among the ordinals of the minimal models of some of the systems discussed in §1. Although the proofs of these latter results will not require any use of the uniformisation theorem, we will find it convenient to use some of the more elementary results and techniques from Jensen's fine-structural theory of L. We thus provide a brief review of the pertinent parts of Jensen's works in §0, where a list of general preliminaries is also furnished.

We remark that some of the techniques which we use in the present paper have been used by us previously in [6] to prove various results about β-models of analysis. Since β-models for analysis are analogous to transitive models for set theory, this is not surprising.

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[3]H. Friedman , Countable models of set theories, Cambridge Summer School in Mathematical Logic, Springer, Berlin, 1973.

[4]K. Gödel , Consistency-proof for the generalized continuum hypothesis, Proceedings of the National Academy of Sciences of the United States of America, vol. 25 (1939), pp. 220224.

[7]R. Gostanian , The next admissible ordinal, Annals of Mathematical Logic, vol. 17 (1979), pp. 133.

[8]R. Jensen , The fine structure of the constructible hierarchy, Annals of Mathematical Logic, vol. 4 (1971), pp. 229308.

[9]R. Jensen and C. Karp , Primitive recursive set functions, Axiomatic Set Theory, Part I, American Mathematical Society, Providence, R. J., 1971.

[10]A. Levy , A hierarchy of formulas of set theory, Memoirs of the American Mathematical Society, no. 57 (1965).

[11]H. Putnam , A note on constructible sets of integers, Notre Dame Journal of Formal Logic, vol. 4 (1962), pp. 270273.

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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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