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

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]**J. Baumgartner, A. Taylor and S. Wagon,*On splitting stationary subsets of large cardinals*, J. Sym Logic**42**(1977), 203-214. MR**58:21619****[Br-Ma]**M. Burke and M. Magidor,*Shelah's pcf theory and its applications*, Ann. Pure and Ap. Logic**50**(1990), 207-254.MR**92f:03053****[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]**M. Foreman,*More saturated ideals*, Cabal Seminar 79-81,Lecture Notes in Math. 1019, Springer, 1983, pp. 1-27. MR**86j:03002****[Fo-Ma-Sh]**M. Foreman, M. Magidor and S. Shelah,*Martin's Maximum, saturated ideals and nonregular ultrafilters,*I, Ann. of Math.**127**(1988), 1-47. MR**89f:03043****[Fo-Wo]**M. Foreman and H. Woodin,*GCH can fail everywhere*, Ann. of Math.**133**(1991), 1-36. MR**91k:03130****[Gi1]**M. Gitik,*The nonstationary ideal on*, Israel Journal of Math. (1984), 257-288. MR**86i:03061****[Gi2]**M. Gitik,*Changing cofinalities and the nonstationary ideal*, Israel Journal of Math.**56**(1986), 280-314. MR**89b:03086****[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]**J. Gregory,*Higher Souslin trees and GCH*, J. Symb. Logic**41**(1976), 663-671. MR**58:5208****[Je-Wo]**T. Jech and H. Woodin,*Saturation of the closed unbounded filter on the set of regular cardinals*, Trans. Am. Math. Soc.**292 (1)**(1985), 345-356. MR**87d:03139****[Je-Ma-Mi-Pr]**T. Jech, M. Magidor, W. Mitchell and K. Prikry,*Precipitous ideals*, J. Symb. Logic**45**(1980), 1-8. MR**81h:03097****[Je-Pr]**T. Jech and K. Prikry,*Ideals of sets and the power set operation*, Bul. Am. Math. Soc.**82**(1976), 593-595. MR**58:21618****[Ki-St]**J. King and C. Steinhorn,*The uniformization property for*, Israel J. Math.**36**(1980), 248-256. MR**82a:03051****[Ku]**K. Kunen,*Saturated ideals*, J. of Sym. Logic**43**(1978), 65-76. MR**80a:03068****[Na]**K. Namba,*On closed unbounded ideal of ordinal numbers*, Comm. Univ. Sancti Pauli**22**(1974), 33-56. MR**50:4308****[Sh1]**S. Shelah,*Proper forcing*, Lec. Notes in Math. 940, Springer-Verlag, 1982. MR**84h:03002****[Sh2]**S. Shelah,*Cardinal Arithmetic*, Oxford Univ. Press, 1994. MR**96e:03001****[Sh3]**S. Shelah,*Appendix: on stationary sets (to ``Classification of nonelementary classes II). Abstract elementary classes") in Proc. of the USA-Israel Conference on Classification theory*, Lec. Notes in Math. 1292 (J. Baldwin, ed.), Springer, 1987, pp. 483-485. MR**91h:03046****[Sh4]**S. Shelah,*On Successors of Singular Cardinals, in Logic Colloquium 78*(M. Boffa, D. van Dalen and K. McAloon, eds.), North Holland, 1979, pp. 357-380. MR**82d:03079****[Sh5]**S. Shelah,*Diamonds and uniformizations*, J. Symb. Logic**49**(1984), 1022-1033. MR**86g:03083****[Sh6]**S. Shelah,*Iterated forcing and normal ideals on*, Is. Jour. of Math.**60**(1987), 345-380. MR**90g:03050****[Sh7]**S. Shelah,*More on stationary coding, in Around Classification Theory of Models*, Lecture Notes in Math.**1182**(1986), 224-246. MR**90a:03037****[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]**R. Solovay,*Real valued measurable cardinals, in Axiomatic Set Theory*, Proceedings of Sym. in Pure Math. XIII**1**(1971), 387-428. MR**45:55****[St]**J. Steel,*The Core Model Iterability Problem*(to appear).**[St-V]**J. Steel and R. Van Wesep,*Two consequence of determinacy consistent with choice*, Trans. An. Math. Soc.**272 (1)**(1982), 67-85. MR**83i:03081****[Wo1]**H. Woodin,*Some consistency results yin ZF using AD*, Cabal Seminar, Lect. Notes in Math. 1019, Springer, 1983, pp. 172-199. MR**86j:03002**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
03E35,
03E40,
04A20

Retrieve articles in all journals with MSC (1991): 03E35, 03E40, 04A20

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