Strongly proper forcing and some problems of Foreman

Abstract

We provide solutions to several problems of Foreman about ideals, several of which are closely related to Mitchell's notion of strongly proper forcing. We prove: (1) Presaturation of a normal ideal implies projective antichain catching, enabling us to provide a solution to a problem from Foreman [8] about ideal projections which is more comprehensive and simpler than the solution obtained in [4]. (2) We solve an older question from Foreman [9] about the relationship between generic hugeness and generic almost hugeness. (3) Finally, we provide solutions to two technical questions from Foreman [7] and [8] related to his Duality Theorem.

Destroying Saturation while Preserving Presaturation at an Inaccessible; an Iterated Forcing Argument

“…If P is δ-proper on a stationary set, then P is δ-presaturated. This fact appears as Fact 2.8 of [4], with proof; their proof, in turn, generalizes a result of Foreman and Magidor in the case of δ = ω 1 (namely, Proposition 3.2 of [9]).…”

“…It remained open as to whether the above could be done for n = 0 and κ an inaccessible cardinal; this was the content of Question 8.5 of [4] and further clarifications provided in [5]. This paper's central result establishes that Question 1.1 is consistently false at κ inaccessible, by an argument analogous to that of Theorem 4.1 of [4]: Theorem 1.2.…”

“…So we will say that an ideal I on κ +n is κ +n+1 -presaturated if I induces a wellfounded generic ultrapower and preserves κ +n+1 . Our template is then: 4], corollary of Theorem 1.2). Any κ +n+1 -presaturated, non-κ +n+1 saturated ideal on κ +n provides a counterexample to Question 1.1.…”

“…To construct such ideals for successor cardinals κ = µ + (with µ regular and mild assumptions on cardinal arithmetic), Cox and Eskew in [4] generalized a forcing of Baumgartner and Taylor in [1] to add a club subset C of κ with < µ-conditions. (Baumgartner and Taylor's original version in [1] was for µ = ω.)…”

“…Remark 1.5. In [4], the analogous theorem (Theorem 4.1(2)) argued that there is an S ∈ I + such that…”