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Transactions of the American Mathematical Society

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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
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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 $ M[G]$ is obtained from a model $ M$ of GCH via any $ \sigma $-finite chain condition notion of forcing (e.g. add Cohen reals or random reals) then in $ M[G]$ every countably complete ideal on $ {\omega _1}$ is $ {\omega _3}$-saturated. If "$ \sigma $-finite chain condition" is weakened to "countable chain condition," then the conclusion no longer holds, but in this case one can conclude that every $ {\omega _2}$-generated countably complete ideal on $ {\omega _1}$ (e.g. the nonstationary ideal) is $ {\omega _3}$-saturated. Some applications to $ {\mathcal{P}_{{\omega _1}}}({\omega _2})$ 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. $ {2^{{\aleph _1}}} > {({2^{{\aleph _0}}})^ + })$), then the consistency of some rather large cardinals becomes both necessary and sufficient.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1982-0645330-7
Article copyright: © Copyright 1982 American Mathematical Society

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