Mathias forcing which does not add dominating reals

Canjar, R. Michael

doi:10.1090/S0002-9939-1988-0969054-2

Mathias forcing which does not add dominating reals
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Proc. Amer. Math. Soc. 104 (1988), 1239-1248 Request permission

Abstract:

Assume that there is no dominating family of reals of cardinality $< c$. We show that there then exists an ultrafilter on the set of natural numbers such that its associated Mathias forcing does not adjoin any real which dominates all ground model reals. Such ultrafilters are necessarily $P$-points with no $Q$-points below them in the Rudin-Keisler order.

References

Andreas Blass, Near coherence of filters. I. Cofinal equivalence of models of arithmetic, Notre Dame J. Formal Logic 27 (1986), no. 4, 579–591. MR 867002, DOI 10.1305/ndjfl/1093636772
Andreas Blass and Saharon Shelah, Near coherence of filters. III. A simplified consistency proof, Notre Dame J. Formal Logic 30 (1989), no. 4, 530–538. MR 1036674, DOI 10.1305/ndjfl/1093635236

Ultrafilters with small generating sets

73

Michael Canjar, Countable ultraproducts without CH, Ann. Pure Appl. Logic 37 (1988), no. 1, 1–79. MR 924678, DOI 10.1016/0168-0072(88)90048-6
W. W. Comfort and S. Negrepontis, The theory of ultrafilters, Die Grundlehren der mathematischen Wissenschaften, Band 211, Springer-Verlag, New York-Heidelberg, 1974. MR 0396267

Propriete de Baire de

muni d’une nouvelle topologie et application a la construction des ultrafiltres

Stephen H. Hechler, A dozen small uncountable cardinals, TOPO 72—general topology and its applications (Proc. Second Pittsburgh Internat. Conf., Pittsburgh, Pa., 1972; dedicated to the memory of Johannes H. de Groot), Lecture Notes in Math., Vol. 378, Springer, Berlin, 1974, pp. 207–218. MR 0369078
Stephen H. Hechler, On a ubiquitous cardinal, Proc. Amer. Math. Soc. 52 (1975), 348–352. MR 380705, DOI 10.1090/S0002-9939-1975-0380705-7
Jussi Ketonen, On the existence of $P$-points in the Stone-Čech compactification of integers, Fund. Math. 92 (1976), no. 2, 91–94. MR 433387, DOI 10.4064/fm-92-2-91-94
A. R. D. Mathias, Happy families, Ann. Math. Logic 12 (1977), no. 1, 59–111. MR 491197, DOI 10.1016/0003-4843(77)90006-7
Arnold W. Miller, There are no $Q$-points in Laver’s model for the Borel conjecture, Proc. Amer. Math. Soc. 78 (1980), no. 1, 103–106. MR 548093, DOI 10.1090/S0002-9939-1980-0548093-2
Christian W. Puritz, Skies, constellations and monads, Contributions to non-standard analysis (Sympos., Oberwolfach, 1970), Studies in Logic and Foundations of Math., Vol. 69, North-Holland, Amsterdam, 1972, pp. 215–243. MR 0645172
Eric K. van Douwen, The integers and topology, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 111–167. MR 776622