More Notions of Forcing Add a Souslin Tree

Transformations of the transfinite plane

“…Proof. By the main result of [BR19b], in V P , an instance of the proxy principle, much stronger than P − µ (κ, 2, µ ⊑, κ, {E κ µ }, 2, 1 1 2 ), holds. So, by Lemma 7.12, if V P |= µ <µ = µ, then V P |= Pℓ 1 (µ + , µ + , µ).…”

Full Souslin trees at small cardinals

“…Fix a sequence $$\vec{C}=\langle C_\alpha \ \lvert \ \alpha &lt;\kappa \rangle$$ witnessing $$\operatorname{P}^-(\kappa,2,{\sqsubseteq ^*},1,\lbrace S\rbrace)$$, and fix a well‐ordering $$\vartriangleleft$$ of $$H_\kappa$$. Following the proof of [2, Proposition 2.2], we shall recursively construct a sequence $$\langle T_\alpha \ \lvert \ \alpha &lt;\kappa \rangle$$ such that $$T:=\bigcup _{\alpha &lt;\kappa }T_\alpha$$ will constitute a normal prolific full streamlined $$\kappa$$‐Souslin tree whose $$\alpha {\text{th}}$$ level is $$T_\alpha$$. Let $$T_0:=\lbrace \emptyset \rbrace$$, and for all $$\alpha &lt;\kappa$$ let …”

“…Proof. We settle for proving Clause(2). So, suppose that ⟨𝑋 𝛽 | 𝛽 ∈ 𝐸 𝜆 𝜃 ⟩ witnesses ♣(𝐸 𝜆 𝜃 ), for a given 𝜃 ∈ Reg(𝜆), and we shall prove that ♢ * 𝑆,𝐵 (𝜅-trees) holds for 𝐵 ∶= 𝐸 𝜅 𝜃 .…”

Souslin trees at successors of regular cardinals

“…As mentioned earlier, even in the presence of GCH, (κ, <κ) does not imply the existence of a κ-Souslin tree. For this, Brodsky and the author have introduced the following slight strengthening of (κ, <κ): (Brodsky & Rinot;[4]) * (κ) asserts the existence of a sequence C α | α < κ such that 1. for every limit ordinal α < κ, (a) C α is a nonempty collection of club subsets of α, with |C α | < κ; and (b) for every C ∈ C α and everyᾱ ∈ acc(C), we have C ∩ᾱ ∈ Cᾱ; 2. for every cofinal X ⊆ κ, there is α ∈ acc(κ) such that sup(nacc(C) ∩ X ) = α for all C ∈ C α .…”