In this paper, we give consistency proofs for the existence of a κ-saturated ideal on an inaccessible κ, and for the existence of an ω2-saturated ideal on ω1. We also include an historical survey outlining other known results on saturated ideals.
§1. Forcing. We assume that the reader is familiar with the usual techniques in forcing and Boolean-valued models (see Jech  or Rosser ), so we shall just specify here some of the less standard notation.
“cBa” abbreviates “complete Boolean algebra”.
If P (= ‹P, <›) is a notion of forcing, is the associated cBa. We write VP for .
Notions like κ-closed, κ-complete, etc. always mean < κ. Thus, P is κ-closed iff every decreasing chain of length less than κ has a lower bound, and a Boolean algebra is κ-complete iff sups of subsets of of cardinality less than κ always exist.
If is a cBa, x̌ is the object in representing x in V. In many cases, especially with ordinals, the ̌ is dropped. V̌ is the Boolean-valued class representing V — i.e.,
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