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 $M[G]$ 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.