Norms on possibilities. I. Forcing with trees and creatures

Sheva-sheva-sheva: Large creatures

“…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.…”

Generating ultrafilters in a reasonable way

“…Put T * ζ = T ζ ∪{η * ↾α : α < ω 1 } and define σ ′′ : <ω1 ω 1 −→ ω 1 so that σ ′′ ↾T * ζ = σ ′ ↾T * ζ and for ν ∈ <ω1 ω 1 \ T * ζ we have (⊛) 3 σ ′′ (ν) = min β < ω 1 : ∃(α, Z, d) ∈ r ζ sup(ν) < α & Z ⊆ β .…”

The splitting number can be smaller than the matrix chaos number

“…Moreover, we explain the connections and give the references to [7], so that the reader can identify it as a special case of an extensive framework. Definition 1.1.…”

“…Hence, after possibly a further deletion of lines we may restrict the set of Toeplitz matrices to linear Toeplitz matrices. A matrix is linear iff each column j has at most one entry a i,j = 0 and for j < j the i with a i,j = 0 is smaller than or equal to the i with a i,j = 0 if both exist, in picture: [7]. In the next two sections we shall show: The c i 's coming from the trunks of the conditions in the generic filter of our forcing Q give matrices that make, after multiplication, members of ω 2 from the ground model and members of ω 2 of any random extension convergent.…”

“…Stronger conditions are obtained by composing (usually finitely many) partial functions and changing the arrangement in a certain way. We need the exact definitions only at one point in our work, when we want to cite some result on properness from [7]. For this purpose, we have to verify that our forcing is an instance of a creature forcing built from a finitary creating pair, such that the forcing conditions fulfil some conditions on the growth of the norms of their building blocks.…”