Iteration of λ‐complete forcing notions not collapsing λ⁺

Abstract

Abstract. We look for a parallel to the notion of "proper forcing" among λ-complete forcing notions not collapsing λ + . We suggest such a definition and prove that it is preserved by suitable iterations.

Sheva-sheva-sheva: Large creatures

“…Strong fuzzy properness is very close to properness over semidiamonds of Ros lanowski and Shelah [15] and even closer to properness over diamonds introduced by Eisworth [6]. (Note that considering the condition A.3.6(⊛) + we may assume that the weak fuzzy candidateq = q δ,x : δ ∈ S is limit & x ∈ X δ is such that |X δ | = 1 for each relevant δ, so one may treat it asq = q δ : δ ∈ S is limit .)…”

“…The next section, A.2, presents preservation of λ-analogue of the Sacks property (in Theorem A.2.3) as well as preservation of being λ λ-bounding (in Theorem A.2.6). Section A.3 introduces fuzzy properness, a more complicated variant of properness over semi-diamonds from [15]. Of course, we prove a suitable iteration theorem (see Theorem A.3.10).…”

“…V]. Then Ros lanowski and Shelah [15] gave an iterable condition for not collapsing λ + in λ-support iterations of λ-complete forcing notions (with possibly adding subsets of λ). Very recently Eisworth [6] has given another property preserved in λ-support iterations (and implying that λ + is not collapsed).…”

“…Very recently Eisworth [6] has given another property preserved in λ-support iterations (and implying that λ + is not collapsed). At the moment it is not clear if the two properties (the one of [15] and that of [6]) are equivalent, though they have similar flavour. However, the existing iterable properties still do not cover many examples of natural forcing notions, specially those which come naturally in the context of λ-reals.…”

“…Our aim in this paper is to provide tools for building forcing notions relevant for λ-reals (continuing in this Ros lanowski and Shelah [14], [13]) and give suitable iteration theorems (thus continuing Ros lanowski and Shelah [15]). However, we restrict our attention to the case when λ is a strongly inaccessible uncountable cardinal (after all, ℵ 0 is inaccessible), see 0.3 below.…”