On the generic existence of special ultrafilters
Author:
R. Michael Canjar
Journal:
Proc. Amer. Math. Soc. 110 (1990), 233241
MSC:
Primary 03E05; Secondary 03E65, 04A20, 54A25
MathSciNet review:
993747
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Abstract: We introduce the concept of the generic existence of point, point, and selective ultrafilters, a concept which is somewhat stronger than the existence of these sorts of ultrafilters. We show that selective ultrafilters exist generically iff semiselectives do iff , and we show that point ultrafilters exist generically iff semipoints do iff , where is the minimal cardinality of a dominating family of functions and is the minimal cardinality of a cover of the real line by nowheredense sets. These results complement a result of Ketonen, that points exist generically iff , and one of P. Nyikos and D. H. Fremlin, that saturated ultrafilters exist generically iff .
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T. Bartoszynski, Combinatorial aspects of measure and category, Univ. Warszawski Inst. Mat., preprint 9/84, 1984. MR 917147 (88m:04001)
 [2]
J. Baumgartner and R. Laver, Iterated perfect set forcing, Ann. Math. Logic 17 (1979), 271288. MR 556894 (81a:03050)
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J. Baumgartner, private letter to the author, 11 November 1988.
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M. Bell and K. Kunen, On the PI character of ultrafilters, C. R. Math. Rep. Acad. Sci. Canada 3 (1981), 351356. MR 642449 (82m:03064)
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A. Blass and S. Shelah, There may be simple and points and the RudinKeisler order may be downward directed, Ann. Pure Appl. Logic 33 (1987), 213243. MR 879489 (88e:03073)
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D. Booth, Ultrafilters on a countable set, Ann. Math. Logic 2 (19701), 124. MR 0277371 (43:3104)
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R. M. Canjar, Countable ultraproducts without CH, Ann. Pure Appl. Logic 37 (1988), 179. MR 924678 (89g:03073)
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, Smallfilter forcing, J. Symbolic Logic 51 (1986), 526546. MR 853837 (87m:03068)
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C. C. Chang and H. J. Keisler, Model theory, NorthHolland, 1973.
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J. Ketonen, On the existence of points, Fund. Math. 92 (1976), 9199. MR 0433387 (55:6363)
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K. Kunen, Some points in ; Proc. Cambridge Philos. Soc. 78 (1980), 385398. MR 0427070 (55:106)
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A. Mathias, Remarks on rare filters, Infinite and Finite Sets, Colloq. Math. Soc. János Bolya, NorthHolland, Amsterdam, 1975. MR 0373898 (51:10098)
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A. Miller, There are no Points in Laver's model for the Borel conjecture, Proc. Amer. Math. Soc. 78 (1980), 103106. MR 548093 (80h:03071)
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, A characterization of the least cardinal for which the Baire category theorem fails, Proc. Amer. Math. Soc. 86 (1982), 498502. MR 671224 (84b:04002)
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P. J. Nyikos and D. H. Fremlin, Saturating ultrafilters on N, J. Symbolic Logic 54 (1989), 708718. MR 1011162 (90i:03050)
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J. Roitman, Nonisomorphic fields from nonisomorophic ultrapowers, Math. Z. 181 (1982), 9396.
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S. Shelah, Proper forcing, Lecture Notes in Math., vol. 940, SpringerVerlag, 1982, pp. 213221. MR 675955 (84h:03002)
 [19]
J. Steprans, Cardinal Arithmetic and Borel sets, Proc. Amer. Math. Soc. 84 (1982), 121126. MR 633292 (83g:03052)
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A. Taylor, On the existence of points and points, preprint.
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E. van Douwen, The integers and topology, Handbook of SetTheoretic Topology (K. Kunen and J. E. Vaughan, eds.), NorthHolland, 1984. MR 776622 (87f:54008)
 [22]
W. Weiss, Versions of Martin's axiom, Handbook of SetTheoretic Topology (K. Kunen and J. E. Vaughan, eds.), NorthHolland, 1984. MR 776638 (86h:03088)
 [1]
 T. Bartoszynski, Combinatorial aspects of measure and category, Univ. Warszawski Inst. Mat., preprint 9/84, 1984. MR 917147 (88m:04001)
 [2]
 J. Baumgartner and R. Laver, Iterated perfect set forcing, Ann. Math. Logic 17 (1979), 271288. MR 556894 (81a:03050)
 [3]
 J. Baumgartner, private letter to the author, 11 November 1988.
 [4]
 M. Bell, On the combinatorial principle , Fund. Math. 114 (1981), 149157. MR 643555 (83e:03077)
 [5]
 M. Bell and K. Kunen, On the PI character of ultrafilters, C. R. Math. Rep. Acad. Sci. Canada 3 (1981), 351356. MR 642449 (82m:03064)
 [6]
 A. Blass and S. Shelah, There may be simple and points and the RudinKeisler order may be downward directed, Ann. Pure Appl. Logic 33 (1987), 213243. MR 879489 (88e:03073)
 [7]
 D. Booth, Ultrafilters on a countable set, Ann. Math. Logic 2 (19701), 124. MR 0277371 (43:3104)
 [8]
 R. M. Canjar, Countable ultraproducts without CH, Ann. Pure Appl. Logic 37 (1988), 179. MR 924678 (89g:03073)
 [9]
 , Smallfilter forcing, J. Symbolic Logic 51 (1986), 526546. MR 853837 (87m:03068)
 [10]
 C. C. Chang and H. J. Keisler, Model theory, NorthHolland, 1973.
 [11]
 J. Ketonen, On the existence of points, Fund. Math. 92 (1976), 9199. MR 0433387 (55:6363)
 [12]
 K. Kunen, Some points in ; Proc. Cambridge Philos. Soc. 78 (1980), 385398. MR 0427070 (55:106)
 [13]
 A. Mathias, Remarks on rare filters, Infinite and Finite Sets, Colloq. Math. Soc. János Bolya, NorthHolland, Amsterdam, 1975. MR 0373898 (51:10098)
 [14]
 A. Miller, There are no Points in Laver's model for the Borel conjecture, Proc. Amer. Math. Soc. 78 (1980), 103106. MR 548093 (80h:03071)
 [15]
 , A characterization of the least cardinal for which the Baire category theorem fails, Proc. Amer. Math. Soc. 86 (1982), 498502. MR 671224 (84b:04002)
 [16]
 P. J. Nyikos and D. H. Fremlin, Saturating ultrafilters on N, J. Symbolic Logic 54 (1989), 708718. MR 1011162 (90i:03050)
 [17]
 J. Roitman, Nonisomorphic fields from nonisomorophic ultrapowers, Math. Z. 181 (1982), 9396.
 [18]
 S. Shelah, Proper forcing, Lecture Notes in Math., vol. 940, SpringerVerlag, 1982, pp. 213221. MR 675955 (84h:03002)
 [19]
 J. Steprans, Cardinal Arithmetic and Borel sets, Proc. Amer. Math. Soc. 84 (1982), 121126. MR 633292 (83g:03052)
 [20]
 A. Taylor, On the existence of points and points, preprint.
 [21]
 E. van Douwen, The integers and topology, Handbook of SetTheoretic Topology (K. Kunen and J. E. Vaughan, eds.), NorthHolland, 1984. MR 776622 (87f:54008)
 [22]
 W. Weiss, Versions of Martin's axiom, Handbook of SetTheoretic Topology (K. Kunen and J. E. Vaughan, eds.), NorthHolland, 1984. MR 776638 (86h:03088)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939199009937473
PII:
S 00029939(1990)09937473
Article copyright:
© Copyright 1990
American Mathematical Society
