Reduced Powers of Souslin Trees

Abstract

We study the relationship between a κ-Souslin tree T and its reduced powers T θ / . Previous works addressed this problem from the viewpoint of a single power θ , whereas here, tools are developed for controlling different powers simultaneously. As a sample corollary, we obtain the consistency of an ℵ 6 -Souslin tree T and a sequence of uniform ultrafilters n | n < 6 such that T ℵn / n is ℵ 6 -Aronszajn if and only if n < 6 is not a prime number. This paper is the first application of the microscopic approach to Souslin-tree construction, recently introduced by the authors. A major component here is devising a method for constructing trees with a prescribed combination of freeness degree and ascent-path characteristics.

Squares, Ascent Paths, and Chain Conditions

“…are defined analogously. (2) We let T C i denote the tree with underlying set S C ≤i and Then c γ is injective on < T -chains in r −1 "{γ}. If α ∈ dom(c γ ) \ A is not minimal in T, then we have acc(C α,i ) = ∅, acc(C α,i ) ∩ ht T (α) = ∅ and therefore ht T (α) ≤ min(acc(C α,i )) = r(α) = γ.…”

A microscopic approach to Souslin-tree constructions, Part I

“…But this relation is also useful in the absence of E λ . For instance, in [BR15], a λ-complete λ-free λ + -Souslin tree was constructed assuming λ <λ = λ and P(λ…”

“…The outcome will always be a tree, but its features depend on which actions where invoked along the way, and to which vector of parameters of the proxy principle the witness was given. Once the construction of Souslin trees becomes so simple, it is then easier to carry out considerably more complex constructions, and this has already been demonstrated in [BR15], where we gave the first example of a Souslin tree whose reduced powers behave independently of each other, proving, e.g., that V = L entails an ℵ 7 -Souslin tree whose reduced ℵ n -power is ℵ 7 -Aronszajn iff n ∈ {0, 1, 4, 5}.…”

“…For example, in [BR15], we constructed a free ℵ ω+1 -Souslin tree in a model of Martin's Maximum (MM), 11 where the sealing of antichains was done at levels α < ℵ ω+1 of countable cofinality, even though MM implies that every stationary subset of E ℵω+1 ω reflects.…”

“…As Equation (1) above makes explicit, the preceding proof did not utilize the full force of the axiom ⊠(κ). For an application of the full force of ⊠(κ), we refer the reader to Section 6 of [BR15].…”