Saturation properties of ideals in generic extensions. II
HTML articles powered by AMS MathViewer
- by James E. Baumgartner and Alan D. Taylor PDF
- Trans. Amer. Math. Soc. 271 (1982), 587-609 Request permission
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
-
B. Balcar and F. Franek, Completion of factor Boolean algebras, handwritten manuscript.
J. Baumgartner, Independence results in set theory, Notices Amer. Math. Soc. 25 (1978), A248-249.
—, Iterated forcing, Proc. 1978 Cambridge Summer School in Logic (to appear).
- James E. Baumgartner and Alan D. Taylor, Saturation properties of ideals in generic extensions. I, Trans. Amer. Math. Soc. 270 (1982), no. 2, 557–574. MR 645330, DOI 10.1090/S0002-9947-1982-0645330-7 J. Baumgartner, A. Taylor and S. Wagon, Structural properties of ideals, Dissertationes Math. (to appear).
- Thomas Jech, Set theory, Pure and Applied Mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 506523
- T. Jech, M. Magidor, W. Mitchell, and K. Prikry, Precipitous ideals, J. Symbolic Logic 45 (1980), no. 1, 1–8. MR 560220, DOI 10.2307/2273349
- Thomas Jech and Karel Prikry, Ideals over uncountable sets: application of almost disjoint functions and generic ultrapowers, Mem. Amer. Math. Soc. 18 (1979), no. 214, iii+71. MR 519927, DOI 10.1090/memo/0214
- Yuzuru Kakuda, On a condition for Cohen extensions which preserve precipitous ideals, J. Symbolic Logic 46 (1981), no. 2, 296–300. MR 613283, DOI 10.2307/2273622
- Kenneth Kunen, Some applications of iterated ultrapowers in set theory, Ann. Math. Logic 1 (1970), 179–227. MR 277346, DOI 10.1016/0003-4843(70)90013-6
- Kenneth Kunen, Saturated ideals, J. Symbolic Logic 43 (1978), no. 1, 65–76. MR 495118, DOI 10.2307/2271949
- Robert M. Solovay, Real-valued measurable cardinals, 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. 397–428. MR 0290961
- Alan D. Taylor, Regularity properties of ideals and ultrafilters, Ann. Math. Logic 16 (1979), no. 1, 33–55. MR 530430, DOI 10.1016/0003-4843(79)90015-9
- Alan D. Taylor, On saturated sets of ideals and Ulam’s problem, Fund. Math. 109 (1980), no. 1, 37–53. MR 594324, DOI 10.4064/fm-109-1-37-53 R. Van Wesep, The non-stationary ideal on ${\omega _1}$ can be ${\omega _2}$-saturated, handwritten manuscript. H. Woodin, An ${\aleph _1}$-dense ${\aleph _1}$-complete ideal on ${\aleph _1}$, handwritten manuscript.
Additional Information
- © Copyright 1982 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 271 (1982), 587-609
- MSC: Primary 03C62; Secondary 03E05, 03E35, 03E40, 03E55
- DOI: https://doi.org/10.1090/S0002-9947-1982-0654852-4
- MathSciNet review: 654852