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Small filter forcing

Published online by Cambridge University Press:  12 March 2014

R. Michael Canjar
R. Michael Canjar*
Affiliation:
Department of Mathematics, Union College, Schenectady, New York 12308
*
Department of Computer Science, Mathematics and Statistics, University of Baltimore, Baltimore, Maryland 21201.
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Qλ is the set of nonprincipal filters on ω which are generated by fewer than λ sets, for λ a fixed uncountable, regular cardinal ≤ c. We analyze forcing with Qλ, where Qλ is partially ordered in such a way that a filter F1 is more informative than F2 iff F1 includes F2. Qλ-forcing adjoins an ultrafilter on ω but adds no new reals. We analyze Qλ-forcing from a forcing-theoretic viewpoint. We also analyze the properties of Qλ-generic ultrafilters. These properties are independent of ZFC and depend very much on the ground model. In particular, we study Qλ-forcing over ground models which are Cohen real extensions, random real extensions, and models which satisfy Martin's Axiom.

In §2 we give notations and definitions, and review some of the basic facts about forcing and ultrafilters which we will use. In §3 we introduce Qλ-forcing and prove some basic lemmas about it. §4 studies Qc-forcing. §§5, 6, and 7 analyze Qλ-forcing over ground models of Martin's Axiom, ground models which are generated by Cohen reals, and ground models which are generated by random reals, respectively. Qλ-forcing over Cohen real and random real models is isomorphic to the notion of forcing which adjoins a Cohen generic subset of λ; this is proved in §8.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 51 , Issue 3 , September 1986 , pp. 526 - 546
Copyright
Copyright © Association for Symbolic Logic 1986

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References

REFERENCES

[BL]Baumgartner, J. E. and Laver, R., Iterated perfect-set forcing, Annals of Mathematical Logic, vol 17(1979), pp. 271–288.CrossRefGoogle Scholar
[BT]Baumgartner, J. E. and Taylor, A. D., Partition theorems and ultrafilters, Transactions of the American Mathematical Society, vol. 241 (1978), pp. 283–309.CrossRefGoogle Scholar
[Bo]Booth, D., Ultrafilters on a countable set, Annals of Mathematical Logic, vol. 2 (1970/1971), pp. 1–24.CrossRefGoogle Scholar
[Ca1]Canjar, M., Model-theoretic properties of countable ultraproducts without the continuum hypothesis, Ph.D. Thesis, University of Michigan, Ann Arbor, Michigan, 1982.Google Scholar
[Ca2]Canjar, M., Countable ultraproducts without CH, Annals of Pure and Applied Logic (to appear).Google Scholar
[CK]Chang, C. C. and Keisler, H. J., Model theory, North-Holland, Amsterdam, 1973.Google Scholar
[ER]Ellentuck, E. and Rucker, R. V. B., Martin's axiom and saturated models, Proceedings of the American Mathematical Society, vol. 34 (1972), pp. 243–249.Google Scholar
[Je]Jech, T., Set theory, Academic Press, New York, 1978.Google Scholar
[Ke]Ketonen, J., On the existence of P-points in the Stone-Čech compactification of integers, Fundamenta Mathematicae, vol. 92 (1976), pp. 91–99.CrossRefGoogle Scholar
[Ku1]Kunen, K., Ultrafilters and independent sets, Transactions of the American Mathematical Society, vol. 172 (1972), pp. 299–306.CrossRefGoogle Scholar
[Ku2]Kunen, K., Some points in βN, Mathematical Proceedings of the Cambridge Philosophical Society, vol. 80(1976), pp. 385–398.CrossRefGoogle Scholar
[Ku3]Kunen, K., Set theory: an introduction to independence proofs, North-Holland, Amsterdam, 1980.Google Scholar
[So]Solovay, R., A model of set-theory in which every set of reals is Lebesgue measurable, Annals of Mathematics, ser. 2, vol. 92 (1970), pp. 1–56.CrossRefGoogle Scholar