Ultrafilters with property
Author:
Arnold W. Miller
Journal:
Proc. Amer. Math. Soc. 137 (2009), 31153121
MSC (2000):
Primary 03E35, 03E17, 03E50
Published electronically:
April 20, 2009
MathSciNet review:
2506470
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: A set has property (s) (Marczewski (Szpilrajn)) iff for every perfect set there exists a perfect set such that or . Suppose is a nonprincipal ultrafilter on . It is not difficult to see that if is preserved by Sacks forcing, i.e., if it generates an ultrafilter in the generic extension after forcing with the partial order of perfect sets, then has property (s) in the ground model. It is known that selective ultrafilters or even Ppoints are preserved by Sacks forcing. On the other hand (answering a question raised by Hrusak) we show that assuming CH (or more generally MA) there exists an ultrafilter with property (s) such that does not generate an ultrafilter in any extension which adds a new subset of .
 1.
Tomek
Bartoszyński and Haim
Judah, Set theory, A K Peters, Ltd., Wellesley, MA, 1995. On
the structure of the real line. MR 1350295
(96k:03002)
 2.
James
E. Baumgartner and Richard
Laver, Iterated perfectset forcing, Ann. Math. Logic
17 (1979), no. 3, 271–288. MR 556894
(81a:03050), http://dx.doi.org/10.1016/00034843(79)90010X
 3.
Andreas
Blass, A partition theorem for perfect
sets, Proc. Amer. Math. Soc.
82 (1981), no. 2,
271–277. MR
609665 (83k:03063), http://dx.doi.org/10.1090/S00029939198106096650
 4.
Jörg
Brendle, Between 𝑃points and nowhere dense
ultrafilters, Israel J. Math. 113 (1999),
205–230. MR 1729447
(2000m:03117), http://dx.doi.org/10.1007/BF02780177
 5.
Alexander
S. Kechris, Classical descriptive set theory, Graduate Texts
in Mathematics, vol. 156, SpringerVerlag, New York, 1995. MR 1321597
(96e:03057)
 6.
Arnold
W. Miller, Rational perfect set forcing, Axiomatic set theory
(Boulder, Colo., 1983) Contemp. Math., vol. 31, Amer. Math. Soc.,
Providence, RI, 1984, pp. 143–159. MR 763899
(86f:03084), http://dx.doi.org/10.1090/conm/031/763899
 7.
Arnold
W. Miller, Special subsets of the real line, Handbook of
settheoretic topology, NorthHolland, Amsterdam, 1984,
pp. 201–233. MR 776624
(86i:54037)
 8.
Jan
Mycielski, Independent sets in topological algebras, Fund.
Math. 55 (1964), 139–147. MR 0173645
(30 #3855)
 9.
Gerald
E. Sacks, Forcing with perfect closed sets, Axiomatic Set
Theory (Proc. Sympos. Pure Math., Vol. XIII, Part I, Univ. California, Los
Angeles, Calif., 1967) Amer. Math. Soc., Providence, R.I., 1971,
pp. 331–355. MR 0276079
(43 #1827)
 10.
Robert
M. Solovay, On the cardinality of ∑₂¹ sets of
reals, Foundations of Mathematics (Symposium Commemorating Kurt
Gödel, Columbus, Ohio, 1966) Springer, New York, 1969,
pp. 58–73. MR 0277382
(43 #3115)
 1.
 Bartoszyński, Tomek; Judah, Haim; Set theory. On the structure of the real line. A K Peters, Ltd., Wellesley, MA, 1995. (Ultrafilters can't be measurable or have property of Baire.) MR 1350295 (96k:03002)
 2.
 Baumgartner, James E.; Laver, Richard; Iterated perfectset forcing. Ann. Math. Logic 17 (1979), no. 3, 271288. MR 556894 (81a:03050)
 3.
 Blass, Andreas; A partition theorem for perfect sets. Proc. Amer. Math. Soc. 82 (1981), no. 2, 271277. MR 609665 (83k:03063)
 4.
 Brendle, Jorg; Between points and nowhere dense ultrafilters. Israel J. Math. 113 (1999), 205230. MR 1729447 (2000m:03117)
 5.
 Kechris, Alexander S.; Classical descriptive set theory. Graduate Texts in Mathematics, 156. SpringerVerlag, New York, 1995. MR 1321597 (96e:03057)
 6.
 Miller, Arnold W.; Rational perfect set forcing. Axiomatic set theory (Boulder, Colo., 1983), 143159, Contemp. Math., 31, Amer. Math. Soc., Providence, RI, 1984. MR 763899 (86f:03084)
 7.
 Miller, Arnold W.; Special subsets of the real line. Handbook of settheoretic topology, 201233, NorthHolland, Amsterdam, 1984. MR 776624 (86i:54037)
 8.
 Mycielski, Jan; Independent sets in topological algebras. Fund. Math. 55 (1964), 139147. MR 0173645 (30:3855)
 9.
 Sacks, Gerald E.; Forcing with perfect closed sets. Axiomatic Set Theory (Proc. Sympos. Pure Math., Vol. XIII, Part I, Univ. California, Los Angeles, Calif., 1967), pp. 331355, Amer. Math. Soc., Providence, RI, 1971. MR 0276079 (43:1827)
 10.
 Solovay, Robert M.; On the cardinality of sets of reals. Foundations of Mathematics (Symposium Commemorating Kurt Gödel, Columbus, Ohio, 1966), pp. 5873, Springer, New York, 1969. MR 0277382 (43:3115)
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (2000):
03E35,
03E17,
03E50
Retrieve articles in all journals
with MSC (2000):
03E35,
03E17,
03E50
Additional Information
Arnold W. Miller
Affiliation:
Department of Mathematics, University of WisconsinMadison, Van Vleck Hall, 480 Lincoln Drive, Madison, Wisconsin 537061388
Email:
miller@math.wisc.edu
DOI:
http://dx.doi.org/10.1090/S0002993909099195
PII:
S 00029939(09)099195
Received by editor(s):
October 27, 2003
Received by editor(s) in revised form:
January 15, 2004
Published electronically:
April 20, 2009
Additional Notes:
Thanks to the Fields Institute, Toronto, for their support during the time these results were proved and to Juris Steprans for helpful conversations, and thanks to Boise State University for support during the time this paper was written
Communicated by:
Alan Dow
Article copyright:
© Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
