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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Saturation properties of ideals in generic extensions. I
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by James E. Baumgartner and Alan D. Taylor PDF
Trans. Amer. Math. Soc. 270 (1982), 557-574 Request permission

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
  • © Copyright 1982 American Mathematical Society
  • 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