Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Saturation properties of ideals in generic extensions. II

Authors: James E. Baumgartner and Alan D. Taylor
Journal: Trans. Amer. Math. Soc. 271 (1982), 587-609
MSC: Primary 03C62; Secondary 03E05, 03E35, 03E40, 03E55
MathSciNet review: 654852
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The general type of problem considered here is the following. Suppose $ I$ is a countably complete ideal on $ {\omega _1}$ satisfying some fairly strong saturation requirement (e.g. $ I$ is precipitous or $ {\omega _2}$-saturated), and suppose that $ P$ is a partial ordering satisfying some kind of chain condition requirement (e.g. $ P$ has the c.c.c. or forcing with $ P$ preserves $ {\omega _1}$). Does it follow that forcing with $ P$ preserves the saturation property of $ I$? In this context we consider not only precipitous and $ {\omega _2}$-saturated ideals, but we also introduce and study a class of ideals that are characterized by a property lying strictly between these two notions. Some generalized versions of Chang's conjecture and Kurepa's hypothesis also arise naturally from these considerations.

References [Enhancements On Off] (What's this?)

  • [BF] B. Balcar and F. Franek, Completion of factor Boolean algebras, handwritten manuscript.
  • [B$ _{1}$] J. Baumgartner, Independence results in set theory, Notices Amer. Math. Soc. 25 (1978), A248-249.
  • [B$ _{2}$] -, Iterated forcing, Proc. 1978 Cambridge Summer School in Logic (to appear).
  • [BT] J. Baumgartner and A. Taylor, Saturation properties of ideals in generic extensions. I, Trans. Amer. Math. Soc. 270 (1982), 557-574. MR 645330 (83k:03040a)
  • [BTW] J. Baumgartner, A. Taylor and S. Wagon, Structural properties of ideals, Dissertationes Math. (to appear).
  • [J] T. Jech, Set theory, Academic Press, New York, 1978. MR 506523 (80a:03062)
  • [JMMP] T. Jech, M. Magidor, W. Mitchell and K. Prikry, Precipitous ideals, J. Symbolic Logic 45 (1980), 1-8. MR 560220 (81h:03097)
  • [JP] T. Jech and K. Prikry, Ideals over uncountable sets: application of almost disjoint functions and generic ultrapowers, Mem. Amer. Math. Soc., No. 214 (1979). MR 519927 (80f:03059)
  • [K] Y. Kakuda, On a condition for Cohen extensions which preserve precipitous ideals (preprint). MR 613283 (82i:03062)
  • [Ku$ _{1}$] K. Kunen, Some applications of iterated ultrapowers in set theory, Ann. Math. Logic 1 (1970), 179-227. MR 0277346 (43:3080)
  • [Ku$ _{2}$] -, Saturated ideals, J. Symbolic Logic 43 (1978), 65-76. MR 495118 (80a:03068)
  • [S] R. Solovay, Real valued measurable cardinals, Axiomatic Set Theory, Proc. Sympos. Pure Math., vol. 13, Amer. Math. Soc., Providence, R. I., 1971, pp. 397-428. MR 0290961 (45:55)
  • [T$ _{1}$] A. Taylor, Regularity properties of ideals and ultrafilters, Ann. Math. Logic 16 (1979), 33-55. MR 530430 (83b:04003)
  • [T$ _{2}$] -, On saturated sets of ideals and Ulam's problem, Fund. Math. 109 (1980), 37-53. MR 594324 (82a:03045)
  • [VW] R. Van Wesep, The non-stationary ideal on $ {\omega _1}$ can be $ {\omega _2}$-saturated, handwritten manuscript.
  • [W] H. Woodin, An $ {\aleph _1}$-dense $ {\aleph _1}$-complete ideal on $ {\aleph _1}$, handwritten manuscript.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 03C62, 03E05, 03E35, 03E40, 03E55

Retrieve articles in all journals with MSC: 03C62, 03E05, 03E35, 03E40, 03E55

Additional Information

Article copyright: © Copyright 1982 American Mathematical Society

American Mathematical Society