Saturation properties of ideals in generic extensions. I

Authors:
James E. Baumgartner and Alan D. Taylor

Journal:
Trans. Amer. Math. Soc. **270** (1982), 557-574

MSC:
Primary 03C62; Secondary 03E05, 03E35, 03E40, 03E55

DOI:
https://doi.org/10.1090/S0002-9947-1982-0645330-7

MathSciNet review:
645330

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We consider saturation properties of ideals in models obtained by forcing with countable chain condition partial orderings. As sample results, we mention the following. If is obtained from a model of GCH via any -finite chain condition notion of forcing (e.g. add Cohen reals or random reals) then in every countably complete ideal on is -saturated. If "-finite chain condition" is weakened to "countable chain condition," then the conclusion no longer holds, but in this case one can conclude that every -generated countably complete ideal on (e.g. the nonstationary ideal) is -saturated. Some applications to are included and the role played by Martin's Axiom is discussed. It is also shown that if these weak saturation requirements are combined with some cardinality constraints (e.g. ), then the consistency of some rather large cardinals becomes both necessary and sufficient.

**[B]**J. Baumgartner,*Almost disjoint sets, the dense set problem and the partition calculus*, Ann. Math. Logic**10**(1976), 401-439. MR**0401472 (53:5299)****[B]**-,*Canonical partition relations*, J. Symbolic Logic**40**(1975), 541-554. MR**0398837 (53:2688)****[BT]**J. Baumgartner and A. Taylor,*Saturation properties of ideals in generic extensions*. II, Trans. Amer. Math. Soc. (to appear). MR**654852 (83k:03040b)****[BTW]**J. Baumgartner, A. Taylor and S. Wagon,*On splitting stationary subsets of large cardinals*, J. Symbolic Logic**42**(1977), 203-214. MR**0505505 (58:21619)****[BTW]**-,*Structural properties of ideals*, Dissertationes Math. (to appear).**[Di]**C. DiPrisco,*Combinatorial properties and supercompact cardinals*, Ph.D. Thesis, M.I.T., 1976.**[DJ]**A. Dodd and R. Jensen,*The core model*, circulated notes (1976).**[EHMR]**P. Erdös, A. Hajnal, A. Máté and R. Rado,*Combinatorial set theory*:*Partition relations for cardinals*(to appear).**[F]**G. Fodor,*Eine Bemerkung zur Theorie der regressiven Funktionen*, Acta Sci. Math. (Szeged)**17**(1956), 139-142. MR**0082450 (18:551d)****[J]**T. Jech,*Some combinatorial problems concerning uncountable cardinals*, Ann. Math. Logic**5**(1973), 165-198. MR**0325397 (48:3744)****[J]**-,*On the number of generators of an ideal*(to appear).**[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)****[Je]**R. Jensen,*Marginalia to a theorem of Silver*, circulated notes (1975).**[K]**Y. Kakuda,*Saturated ideals in Boolean extensions*, Nagoya Math. J.**48**(1972), 159-168. MR**0316248 (47:4796)****[Ka]**A. Kanamori,*Perfect set forcing for uncountable cardinals*, Ann. Math. Logic**19**(1980), 97-114. MR**593029 (82i:03061)****[Ke]**J. Ketonen,*Some combinatorial principles*, Trans. Amer. Math. Soc.**188**(1974), 387-394. MR**0332481 (48:10808)****[Ku]**D. Kueker,*Countable approximations and Löwenheim-Skolem theorems*, Ann. Math. Logic**11**(1977), 57-103. MR**0457191 (56:15406)****[Kun]**K. Kunen,*Saturated ideals*, J. Symbolic Logic**43**(1978), 65-76. MR**495118 (80a:03068)****[KP]**K. Kunen and J. Paris,*Boolean extensions and measurable cardinals*, Ann. Math. Logic**2**(1971), 359-378. MR**0277381 (43:3114)****[L]**R. Laver,*Making the supercompactness of**indestructible under**directed closed forcing*, Israel J. Math.**29**(1978), 385-388. MR**0472529 (57:12226)****[M]**M. Magidor,*Changing cofinality of cardinals*, Fund. Math.**159**(1978), 61-71. MR**0465868 (57:5754)****[M]**-,*On the singular cardinals problem*. II, Ann. of Math. (to appear).**[Me]**T. Menas,*On strong compactness and supercompactness*, Ann. Math. Logic**7**(1975), 327-359. MR**0357121 (50:9589)****[Me]**-,*Consistency results concerning supercompactness*, Trans. Amer. Math. Soc.**223**(1976), 61-91. MR**0540771 (58:27488)****[MR]**E. Milnor and R. Rado,*The pigeon-hole principle for ordinal numbers*, Proc. London Math. Soc.**15**(1965), 750-768. MR**0190003 (32:7419)****[P]**K. Prikry,*Changing measurable into accessible cardinals*, Dissertationes Math.**68**(1970), 5-52. MR**0262075 (41:6685)****[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]**A. Taylor,*Regularity properties of ideals and ultrafilters*, Ann. Math. Logic**16**(1979), 33-55. MR**530430 (83b:04003)****[T]**-,*On saturated sets of ideals and Ulam's problem*, Fund. Math.**109**(1980), 37-53. MR**594324 (82a:03045)****[W]**S. Wagon,*The saturation of a product of ideals*, Canad. J. Math.**32**(1980), 70-75. MR**559787 (81b:03058)**

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

DOI:
https://doi.org/10.1090/S0002-9947-1982-0645330-7

Article copyright:
© Copyright 1982
American Mathematical Society