Saturation Properties of Ideals in Generic Extensions

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 a-finite chain condition notion of forcing (e.g. add Cohen reals or random reals) then in M[G] every countably complete ideal on (2K°)+), then the consistency of some rather large cardinals becomes both necessary and sufficient. 0. Introduction. Suppose 7 is a K-complete (proper) ideal on k and P is a notion of forcing satisfying the K-chain condition. Then it is well known that 7 generates a (proper) K-complete ideal on k in the generic extension obtained by forcing with P. The results we obtain in this paper and its sequel [BT] are motivated by (but not limited to) the following type of question. For a given cardinal X, what properties of / and P guarantee that this ideal generated by / in the generic extension is X-saturated? Considerations of this type of question are not new to the literature. For example, it is shown in [B,] that one cannot always obtain 2Nl almost-disjoint subsets of <o, of size X,. The proof of this result (which we will greatly generalize in what follows) actually shows that if I — Iu (the ideal of countable subsets of <o,), and X = (2*°)+ as computed in the ground model, then the answer to the question in paragraph one is "none". Hence, one obtains the consistency of 7 being w3-saturated while 2K| is arbitrarily large by simply blowing up the continuum in any model of CH via any c.c.c. notion of forcing. Another example of this type of result is the well-known theorem of Prikry [P] asserting that if X < k and 7 is a X-saturated K-complete ideal on k, then 7 generates a X-saturated K-complete ideal on k in any X-c.c. generic extension. Hence, for example, if k is measurable and one blows up the continuum to k via any c.c.c. notion of forcing, then one obtains a model in which there is an w,-saturated Received by the editors October 3, 1979 and, in revised form, May 26, 1980. 1980 Mathematics Subject Classification. Primary 03E35, 03E05, 04A20. 1 Research partially supported by NSF grant MCS 76-08231. 2 Research partially supported by NSF grant MCS 77-04147. ©1982 American Mathematical Society 0O02-9947/81/O00O-1003/$05.50