Saturation properties of ideals in generic extensions. I

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 ] M[G] is obtained from a model M M of GCH via any σ \sigma -finite chain condition notion of forcing (e.g. add Cohen reals or random reals) then in M [ G ] M[G] every countably complete ideal on ω 1 {\omega _1} is ω 3 {\omega _3} -saturated. If " σ \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 ω 2 {\omega _2} -generated countably complete ideal on ω 1 {\omega _1} (e.g. the nonstationary ideal) is ω 3 {\omega _3} -saturated. Some applications to P ω 1 ( ω 2 ) {\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 ℵ 1 > ( 2 ℵ 0 ) + ) {2^{{\aleph _1}}} > {({2^{{\aleph _0}}})^ + }) ), then the consistency of some rather large cardinals becomes both necessary and sufficient.