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The model of set theory generated by countably many generic reals

Published online by Cambridge University Press:  12 March 2014

Andreas Blass
Andreas Blass*
Affiliation:
University of Michigan, Ann Arbor, Michigan 48109
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Abstract

Adjoin, to a countable standard model M of Zermelo-Fraenkel set theory (ZF), a countable set A of independent Cohen generic reals. If one attempts to construct the model generated over M by these reals (not necessarily containing A as an element) as the intersection of all standard models that include M ∪ A, the resulting model fails to satisfy the power set axiom, although it does satisfy all the other ZF axioms. Thus, there is no smallest ZF model including M ∪ A, but there are minimal such models. These are classified by their sets of reals, and there is one minimal model whose set of reals is the smallest possible. We give several characterizations of this model, we determine which weak axioms of choice it satisfies, and we show that some better known models are forcing extensions of it.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 46 , Issue 4 , December 1981 , pp. 732 - 752
Copyright
Copyright © Association for Symbolic Logic 1981

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References

REFERENCES

[1]Cohen, P. J., Set theory and the continuum hypothesis, Benjamin, New York, 1966.Google Scholar
[2]Feferman, S., Some applications of the notions of forcing and generic sets, Fundamenta Mathematicae, vol. 56 (1965), pp. 325–345.CrossRefGoogle Scholar
[3]Grigorieff, S., Intermediate submodels and generic extensions in set theory, Annals of Mathematics, vol. 101 (1975), pp. 447–490.CrossRefGoogle Scholar
[4]Halpern, J. D. and Lévy, A., The Boolean prime ideal theorem does not imply the axiom of choice, Axiomatic set theory (Scott, D., Editor), Proceedings of Symposia in Pure Mathematics, vol. XIII, part 1, American Mathematical Society, Providence, RI, 1971, pp. 83–134.Google Scholar
[5]Jech, T., The axiom of choice, North-Holland, Amsterdam, 1973.Google Scholar
[6]Jech, T., Set theory, Academic Press, New York, 1978.Google Scholar
[7]Myhill, J. and Scott, D., Ordinal definability, Axiomatic set theory (Scott, D., Editor), Proceedings of Symposia in Pure Mathematics, vol. XIII, part 1, American Mathematical Society, Providence, RI, 1971, pp. 271–278.CrossRefGoogle Scholar
[8]Pincus, D., The strength of the Hahn-Banach theorem, Victoria Symposium on Nonstandard Analysis (Hurd, A. and Loeb, P., Editors), Lecture Notes in Mathematics, no. 369, Springer-Verlag, Berlin and New York, 1974, pp. 203–248.CrossRefGoogle Scholar
[9]Shoenfield, J., Unramified forcing, Axiomatic set theory (Scott, D., Editor), Proceedings of Symposia in Pure Mathematics, vol. XIII, part 1, American Mathematical Society, Providence, RI, 1971, pp. 357–382.CrossRefGoogle Scholar
[10]Solovay, R., A model of set theory in which every set of reals is Lebesgue measurable, Annals of Mathematics, vol. 92 (1970), pp. 1–56.CrossRefGoogle Scholar
[11]Solovay, R. and Tennenbaum, S., Iterated Cohen extensions and Souslin's problem, Annals of Mathematics, vol. 94 (1971), pp. 201–245.CrossRefGoogle Scholar
[12]Vopěnka, P. and Hájek, P., The theory of semisets, North-Holland, Amsterdam, 1972.Google Scholar