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

Article copyright:
© Copyright 1982
American Mathematical Society