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The pure part of HYP(ℳ)

Published online by Cambridge University Press:  12 March 2014

Mark Nadel  and
Jonathan Stavi
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Abstract

Let ℳ be a structure for a language ℒ on a set M of urelements. HYP(ℳ) is the least admissible set above ℳ. In §1 we show that pp(HYP(ℳ)) [= the collection of pure sets in HYP(ℳ)] is determined in a simple way by the ordinal α = ° (HYP(ℳ)) and the ℒxω theory of ℳ up to quantifier rank α. In §2 we consider the question of which pure countable admissible sets are of the form pp(HYP(ℳ)) for some ℳ and show that all sets Lα (α admissible) are of this form. Other positive and negative results on this question are obtained.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 42 , Issue 1 , March 1977 , pp. 33 - 46
Copyright
Copyright © Association for Symbolic Logic 1977

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References

REFERENCES

[1]Barwise, J., Admissible sets and structures, Springer-Verlag, Berlin and New York, 1975.CrossRefGoogle Scholar
[2]Friedman, H., Countable models of set theories, Lecture Notes in Mathematics, vol. 337, Springer-Verlag, Berlin and New York, pp. 539–573.Google Scholar
[3]Guzicki, W., Uncountable β-models with countable height, Fundamenta Mathematicae, vol. 82 (1974–1975), pp. 143–152.CrossRefGoogle Scholar
[4]Karp, C., Finite quantifier equivalence, The theory of models (Addison, , Henkln, , Tarski, , Editors), North-Holland, Amsterdam, 1965, pp. 407–412.Google Scholar
[5]Kueker, D. W., Definability, automorphisms and infinitary languages, Lectures Notes in Mathematics, vol. 72, Springer-Verlag, Berlin and New York, 1968, pp. 152–165.Google Scholar
[6]Lévy, A., A hierarchy of formulas in set theory, Memoirs of the American Mathematical Society, no. 57, (1965), 76 pp.Google Scholar
[7]Nadel, M., More Löwenheim-Skolem results for admissible sets, Israel Journal of Mathematics, vol. 18 (1974), pp. 53–64.CrossRefGoogle Scholar
[8]Nadel, M., Scott sentences and admissible sets, Annals of Mathematical Logic, vol. 7 (1974), pp. 267–294.CrossRefGoogle Scholar
[9]Schlipf, J., Some hyperelementary aspects of model theory, Ph.D. Thesis, University of Wisconsin, 05 1975.Google Scholar
[10]Simpson, S., Subsystems of analysis, unpublished lecture notes, Berkeley, 1973.Google Scholar