Saturation properties of ideals in generic extensions. I
Authors:
James E. Baumgartner and Alan D. Taylor
Journal:
Trans. Amer. Math. Soc. 270 (1982), 557574
MSC:
Primary 03C62; Secondary 03E05, 03E35, 03E40, 03E55
MathSciNet review:
645330
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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.
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 [Di]
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 A. Dodd and R. Jensen, The core model, circulated notes (1976).
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 G. Fodor, Eine Bemerkung zur Theorie der regressiven Funktionen, Acta Sci. Math. (Szeged) 17 (1956), 139142. MR 0082450 (18:551d)
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 T. Jech, Some combinatorial problems concerning uncountable cardinals, Ann. Math. Logic 5 (1973), 165198. MR 0325397 (48:3744)
 [J]
 , On the number of generators of an ideal (to appear).
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 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)
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 R. Jensen, Marginalia to a theorem of Silver, circulated notes (1975).
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 Y. Kakuda, Saturated ideals in Boolean extensions, Nagoya Math. J. 48 (1972), 159168. MR 0316248 (47:4796)
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 A. Kanamori, Perfect set forcing for uncountable cardinals, Ann. Math. Logic 19 (1980), 97114. MR 593029 (82i:03061)
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 J. Ketonen, Some combinatorial principles, Trans. Amer. Math. Soc. 188 (1974), 387394. MR 0332481 (48:10808)
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 D. Kueker, Countable approximations and LöwenheimSkolem theorems, Ann. Math. Logic 11 (1977), 57103. MR 0457191 (56:15406)
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 K. Kunen, Saturated ideals, J. Symbolic Logic 43 (1978), 6576. MR 495118 (80a:03068)
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 K. Kunen and J. Paris, Boolean extensions and measurable cardinals, Ann. Math. Logic 2 (1971), 359378. MR 0277381 (43:3114)
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 R. Laver, Making the supercompactness of indestructible under directed closed forcing, Israel J. Math. 29 (1978), 385388. MR 0472529 (57:12226)
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 M. Magidor, Changing cofinality of cardinals, Fund. Math. 159 (1978), 6171. MR 0465868 (57:5754)
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 , Consistency results concerning supercompactness, Trans. Amer. Math. Soc. 223 (1976), 6191. MR 0540771 (58:27488)
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 E. Milnor and R. Rado, The pigeonhole principle for ordinal numbers, Proc. London Math. Soc. 15 (1965), 750768. MR 0190003 (32:7419)
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 K. Prikry, Changing measurable into accessible cardinals, Dissertationes Math. 68 (1970), 552. MR 0262075 (41:6685)
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 A. Taylor, Regularity properties of ideals and ultrafilters, Ann. Math. Logic 16 (1979), 3355. MR 530430 (83b:04003)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198206453307
PII:
S 00029947(1982)06453307
Article copyright:
© Copyright 1982
American Mathematical Society
