Less saturated ideals

Gitik, Moti; Shelah, Saharon

doi:10.1090/S0002-9939-97-03702-7

Less saturated ideals
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by Moti Gitik and Saharon Shelah

Proc. Amer. Math. Soc. 125 (1997), 1523-1530

DOI: https://doi.org/10.1090/S0002-9939-97-03702-7

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Abstract:

We prove the following:

[(1)] If $\kappa$ is weakly inaccessible then $NS_{\kappa }$ is not $\kappa ^{+}$-saturated.

[(2)] If $\kappa$ is weakly inaccessible and $\theta < \kappa$ is regular then $NS^{\theta }_{\kappa }$ is not $\kappa ^{+}$-saturated.

[(3)] If $\kappa$ is singular then $NS^{cf \kappa }_{ \kappa ^{+}}$ is not $\kappa ^{++}$-saturated.

Combining this with previous results of Shelah, one obtains the following:

[(A)] If $\kappa >\aleph _{1}$ then $NS_{\kappa }$ is not $\kappa ^{+}$-saturated.

[(B)] If $\theta ^{+}< \kappa$ then $NS^{\theta }_{\kappa }$ is not $\kappa ^{+}$-saturated.

References

Olga Taussky, An algebraic property of Laplace’s differential equation, Quart. J. Math. Oxford Ser. 10 (1939), 99–103. MR 83, DOI 10.1093/qmath/os-10.1.99
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, DOI 10.2307/2272121
Maxim R. Burke and Menachem Magidor, Shelah’s $\textrm {pcf}$ theory and its applications, Ann. Pure Appl. Logic 50 (1990), no. 3, 207–254. MR 1086455, DOI 10.1016/0168-0072(90)90057-9
J. Cummings and H. Woodin, Generalized Prikry Forcing (to appear).
M. Džamonja and S. Shelah, On squares, outside guessing of clubs and $I_{<f}[\lambda ]$, Fund. Math. 148 (1995), 165–198. .
M. Džamonja and S. Shelah, Saturated filters at filters at successors of singulars, weak reflection (to appear; Shelah’s paper 545).
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, DOI 10.1007/BFb0071690
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, DOI 10.2307/1971415
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, DOI 10.2307/2944324
Moti Gitik, The nonstationary ideal on $\aleph _2$, Israel J. Math. 48 (1984), no. 4, 257–288. MR 776310, DOI 10.1007/BF02760629
Moti Gitik, Changing cofinalities and the nonstationary ideal, Israel J. Math. 56 (1986), no. 3, 280–314. MR 882254, DOI 10.1007/BF02782938
M. Gitik, Some results on the nonstationary ideal, Israel Journal of Math. 92 (1995), 61-112.
M. Gitik, Some results on nonstationary ideal 2, to appear, Israel J. of Math.
John Gregory, Higher Souslin trees and the generalized continuum hypothesis, J. Symbolic Logic 41 (1976), no. 3, 663–671. MR 485361, DOI 10.2307/2272043
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, DOI 10.1090/S0002-9947-1985-0805967-1
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, On ideals of sets and the power set operation, Bull. Amer. Math. Soc. 82 (1976), no. 4, 593–595. MR 505504, DOI 10.1090/S0002-9904-1976-14121-3
Charles I. Steinhorn and James H. King, The uniformization property for $\aleph _{2}$, Israel J. Math. 36 (1980), no. 3-4, 248–256. MR 597452, DOI 10.1007/BF02762048
Kenneth Kunen, Saturated ideals, J. Symbolic Logic 43 (1978), no. 1, 65–76. MR 495118, DOI 10.2307/2271949
Kanji Namba, On the closed unbounded ideal of ordinal numbers, Comment. Math. Univ. St. Paul. 22 (1973/74), no. 2, 33–56. MR 351820
Saharon Shelah, Proper forcing, Lecture Notes in Mathematics, vol. 940, Springer-Verlag, Berlin-New York, 1982. MR 675955
Saharon Shelah, Cardinal arithmetic, Oxford Logic Guides, vol. 29, The Clarendon Press, Oxford University Press, New York, 1994. Oxford Science Publications. MR 1318912
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, DOI 10.1007/BFb0082243
Saharon Shelah, On successors of singular cardinals, Logic Colloquium ’78 (Mons, 1978) Studies in Logic and the Foundations of Mathematics, vol. 97, North-Holland, Amsterdam-New York, 1979, pp. 357–380. MR 567680
Saharon Shelah, Diamonds, uniformization, J. Symbolic Logic 49 (1984), no. 4, 1022–1033. MR 771774, DOI 10.2307/2274258
Saharon Shelah, Iterated forcing and normal ideals on $\omega _1$, Israel J. Math. 60 (1987), no. 3, 345–380. MR 937796, DOI 10.1007/BF02780398
Saharon Shelah, Around classification theory of models, Lecture Notes in Mathematics, vol. 1182, Springer-Verlag, Berlin, 1986. MR 850051, DOI 10.1007/BFb0098503
S. Shelah, Non structure theory, Oxford Univ. Press ([Sh-h]) (to appear).
S. Shelah, The Generalized Continuum Hypothesis Revisited ([Sh460]) . (to appear).
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
J. Steel, The Core Model Iterability Problem (to appear).
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, DOI 10.1090/S0002-9947-1982-0656481-5
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, DOI 10.1007/BFb0071690