Less Saturated Ideals
Authors:
Moti Gitik and Saharon Shelah
Journal:
Proc. Amer. Math. Soc. 125 (1997), 1523-1530
MSC (1991):
Primary 03E35, 03E40, 04A20
DOI:
https://doi.org/10.1090/S0002-9939-97-03702-7
MathSciNet review:
1363421
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: We prove the following:
- (1)
- If
is weakly inaccessible then
is not
-saturated.
- (2)
- If
is weakly inaccessible and
is regular then
is not
-saturated.
- (3)
- If
is singular then
is not
-saturated.
- (A)
- If
then
is not
-saturated.
- (B)
- If
then
is not
-saturated.
- [Ba-Ta] J. Baumgartner and A. Taylor, Saturation properties of ideals in generic extensions I, II, Trans. Am. Math Soc. 270 (1982), 557-574 and 271 (1982), 587-609. MR 83k:03040a,b
- [Ba-Ta-Wa] James E. Baumgartner, Alan D. Taylor, and Stanley Wagon, On splitting stationary subsets of large cardinals, J. Symbolic Logic 42 (1977), no. 2, 203–214. MR 505505, https://doi.org/10.2307/2272121
- [Br-Ma] Maxim R. Burke and Menachem Magidor, Shelah’s 𝑝𝑐𝑓 theory and its applications, Ann. Pure Appl. Logic 50 (1990), no. 3, 207–254. MR 1086455, https://doi.org/10.1016/0168-0072(90)90057-9
- [Cu-Wo] J. Cummings and H. Woodin, Generalized Prikry Forcing (to appear).
- [Dz-Sh]
M. Dzamonja and S. Shelah, On squares, outside guessing of clubs and
, Fund. Math. 148 (1995), 165-198. CMP 96:04.
- [Dz-Sh2] M. Dzamonja and S. Shelah, Saturated filters at filters at successors of singulars, weak reflection (to appear; Shelah's paper 545).
- [Fo] A. S. Kechris, D. A. Martin, and Y. N. Moschovakis (eds.), Cabal seminar 79–81, Lecture Notes in Mathematics, vol. 1019, Springer-Verlag, Berlin, 1983. MR 730583
- [Fo-Ma-Sh] M. Foreman, M. Magidor, and S. Shelah, Martin’s maximum, saturated ideals, and nonregular ultrafilters. I, Ann. of Math. (2) 127 (1988), no. 1, 1–47. MR 924672, https://doi.org/10.2307/1971415
- [Fo-Wo] Matthew Foreman and W. Hugh Woodin, The generalized continuum hypothesis can fail everywhere, Ann. of Math. (2) 133 (1991), no. 1, 1–35. MR 1087344, https://doi.org/10.2307/2944324
- [Gi1] Moti Gitik, The nonstationary ideal on ℵ₂, Israel J. Math. 48 (1984), no. 4, 257–288. MR 776310, https://doi.org/10.1007/BF02760629
- [Gi2] Moti Gitik, Changing cofinalities and the nonstationary ideal, Israel J. Math. 56 (1986), no. 3, 280–314. MR 882254, https://doi.org/10.1007/BF02782938
- [Gi3] M. Gitik, Some results on the nonstationary ideal, Israel Journal of Math. 92 (1995), 61-112. CMP 96:03
- [Gi4] M. Gitik, Some results on nonstationary ideal 2, to appear, Israel J. of Math.
- [Gr] John Gregory, Higher Souslin trees and the generalized continuum hypothesis, J. Symbolic Logic 41 (1976), no. 3, 663–671. MR 485361, https://doi.org/10.2307/2272043
- [Je-Wo] Thomas J. Jech and W. Hugh Woodin, Saturation of the closed unbounded filter on the set of regular cardinals, Trans. Amer. Math. Soc. 292 (1985), no. 1, 345–356. MR 805967, https://doi.org/10.1090/S0002-9947-1985-0805967-1
- [Je-Ma-Mi-Pr] T. Jech, M. Magidor, W. Mitchell, and K. Prikry, Precipitous ideals, J. Symbolic Logic 45 (1980), no. 1, 1–8. MR 560220, https://doi.org/10.2307/2273349
- [Je-Pr] Thomas Jech and Karel Prikry, On ideals of sets and the power set operation, Bull. Amer. Math. Soc. 82 (1976), no. 4, 593–595. MR 505504, https://doi.org/10.1090/S0002-9904-1976-14121-3
- [Ki-St] Charles I. Steinhorn and James H. King, The uniformization property for ℵ₂, Israel J. Math. 36 (1980), no. 3-4, 248–256. MR 597452, https://doi.org/10.1007/BF02762048
- [Ku] Kenneth Kunen, Saturated ideals, J. Symbolic Logic 43 (1978), no. 1, 65–76. MR 495118, https://doi.org/10.2307/2271949
- [Na] Kanji Namba, On the closed unbounded ideal of ordinal numbers, Comment. Math. Univ. St. Paul. 22 (1973/74), no. 2, 33–56. MR 351820
- [Sh1] Saharon Shelah, Proper forcing, Lecture Notes in Mathematics, vol. 940, Springer-Verlag, Berlin-New York, 1982. MR 675955
- [Sh2] Saharon Shelah, Cardinal arithmetic, Oxford Logic Guides, vol. 29, The Clarendon Press, Oxford University Press, New York, 1994. Oxford Science Publications. MR 1318912
- [Sh3] Saharon Shelah, Classification of nonelementary classes. II. Abstract elementary classes, Classification theory (Chicago, IL, 1985) Lecture Notes in Math., vol. 1292, Springer, Berlin, 1987, pp. 419–497. MR 1033034, https://doi.org/10.1007/BFb0082243
- [Sh4] Saharon Shelah, On successors of singular cardinals, Logic Colloquium ’78 (Mons, 1978) Stud. Logic Foundations Math., vol. 97, North-Holland, Amsterdam-New York, 1979, pp. 357–380. MR 567680
- [Sh5] Saharon Shelah, Diamonds, uniformization, J. Symbolic Logic 49 (1984), no. 4, 1022–1033. MR 771774, https://doi.org/10.2307/2274258
- [Sh6] Saharon Shelah, Iterated forcing and normal ideals on 𝜔₁, Israel J. Math. 60 (1987), no. 3, 345–380. MR 937796, https://doi.org/10.1007/BF02780398
- [Sh7] Saharon Shelah, Around classification theory of models, Lecture Notes in Mathematics, vol. 1182, Springer-Verlag, Berlin, 1986. MR 850051
- [Sh8] S. Shelah, Non structure theory, Oxford Univ. Press ([Sh-h]) (to appear).
- [Sh9] S. Shelah, The Generalized Continuum Hypothesis Revisited ([Sh460]) . (to appear).
- [So] 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
- [St] J. Steel, The Core Model Iterability Problem (to appear).
- [St-V] John R. Steel and Robert Van Wesep, Two consequences of determinacy consistent with choice, Trans. Amer. Math. Soc. 272 (1982), no. 1, 67–85. MR 656481, https://doi.org/10.1090/S0002-9947-1982-0656481-5
- [Wo1] A. S. Kechris, D. A. Martin, and Y. N. Moschovakis (eds.), Cabal seminar 79–81, Lecture Notes in Mathematics, vol. 1019, Springer-Verlag, Berlin, 1983. MR 730583
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Additional Information
Moti Gitik
Affiliation:
School of Mathematical Sciences, Sackler Faculty of Exact Sciences, Tel Aviv University, Ramat Aviv 69978, Israel
Email:
gitik@math.tau.ac.il
Saharon Shelah
Affiliation:
Hebrew University of Jerusalem, Department of Mathematics, Givat Ram, Jerusalem, Israel
Email:
shelah@math.huji.ac.il
DOI:
https://doi.org/10.1090/S0002-9939-97-03702-7
Received by editor(s):
March 1, 1995
Received by editor(s) in revised form:
November 20, 1995
Additional Notes:
The second author was partially supported by the Basic Research Fund, Israel Academy of Sciences.
This is number 577 in the cumulative list of the second author’s publications.
Communicated by:
Andreas R. Blass
Article copyright:
© Copyright 1997
American Mathematical Society