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A microscopic approach to Souslin-tree construction. II. (English) Zbl 1477.03177

In [A. M. Brodsky and A. Rinot, Ann. Pure Appl. Logic 168, No. 11, 1949–2007 (2017; Zbl 1422.03093)] the authors introduced a combinatorial principle \(P^-(\kappa,\mu,{\mathcal R},\theta,{\mathcal S},\nu,\sigma)\) called proxy principle. It establishes the existence of a sequence \(\langle{\mathcal C}_\alpha:\alpha<\kappa\rangle\), with each \(\mathcal C_\alpha\) being a set of clubs on \(\alpha\). Its seven parameters allow for great generality, so some instances of the principle are equivalent to well-known combinatorial axioms, while others are their strengthenings or weakenings. Consistency strength of these instances was investigated, and several constructions of Souslin trees were performed, which was the main motivation for the introduction of proxy principle in the first place. Several more papers by the same authors followed, with various applications.
This is a sequel to the above-mentioned paper. Beside \(P^-\), two more principles are introduced: \(P_\xi^-\) (with an additional argument \(\xi\), bounding the order type of \(C\in{\mathcal C}_\alpha\)) and \(P_\xi^\bullet\) (adding the assumption \(\kappa^{<\kappa}=\kappa\) to \(P_\xi^-\), thus weakening the diamond that is often used in constructions). Sections 4 and 5 are devoted to the analysis of various instances of these principles, their mutual implications, as well as their relationships to known combinatorial principles. In many cases, a proxy principle is strictly weaker than some “old” principle, but still sufficient for the construction of a wanted tree. In addition, this general approach allows some of the results also to hold for a broader class of cardinals \(\kappa\). An example is the following result, the conclusion of which suffices for a construction of a \(\kappa\)-Souslin tree.
Theorem 4.28. Suppose that \(\kappa\) is a strongly inaccessible cardinal and that there exists a sequence \(\langle A_\alpha:\alpha\in S\rangle\) such that:
(i) \(S\) is a nonreflecting stationary subset of \(E_{>\omega}^\kappa\);
(ii) for every \(\alpha\in S\), \(A_\alpha\) is a cofinal subset of \(\alpha\), and
(iii) for every cofinal \(B\subseteq\kappa\), there exists \(\alpha\in S\) for which \(\{\delta<\alpha:\min(A_\alpha\setminus(\delta+1))\in B\}\) is stationary in \(\alpha\).
Then \(P^-(\kappa,\kappa,{\;}^S\!\sqsubseteq,1,\{S\},2,<\omega)\) holds.
Section 6 presents several constructions of \(\kappa\)-Souslin trees with various additional properties. For example, the following theorem is used as a ”prototype construction”.
Theorem 6.8. Suppose that \(\kappa\) is \((<\chi)\)-closed for a given \(\chi\in\mathrm{Reg}(\kappa)\). Let \(\zeta<\kappa\). If \(P^\bullet(\kappa,\kappa,{\;}_\chi\!\sqsubseteq^*,1,\{E_{\geq\chi}^\kappa\},\kappa,<\omega)\) holds, then there exists a normal, prolific, \(\zeta\)-splitting, \(\chi\)-complete \(\kappa\)-Souslin tree.
Theorem 6.11 adds another condition for the constructed tree: that it admits a \(\mu\)-ascending path. When translated into known combinatorial principles, the next improvement of an old result of Baumgartner is obtained.
Corollary 6.12. Suppose that \(\lambda\) is an uncountable cardinal, \(2^\lambda=\lambda^+\) and \(\Box(\lambda^+)\) holds. For every \(\mu\in\mathrm{Reg}(\lambda)\) satisfying \(\lambda^\mu=\lambda\), there exists a \(\lambda^+\)-Souslin tree admitting a \(\mu\)-ascent path.
It is worth mentioning that a separate section (Section 2) is written as a guide “how to construct a \(\kappa\)-Souslin tree the right way”. A thorough analysis was carried of one such construction, with \(\diamondsuit(\kappa)\) and \(\boxtimes^-(\kappa)\) (an instance of proxy) as assumptions. What “microscopic approach” means is the following. When choosing which branches from \(T\upharpoonright\alpha\) will be extended to the level \(T_\alpha\) of the tree, one wants to pick such a branch \(b_x\) for every \(x\in T\upharpoonright\alpha\) (this is important in order to “seal” possible antichains in \(T\) – confine them to being bounded within the first \(\alpha\) levels). It is possible to choose these branches coherently, so that their elements above \(x\) do not depend on \(\alpha\). This enables the set of levels on which antichains are sealed to be a reflecting stationary set, which was not the case in earlier constructions.

MSC:

03E05 Other combinatorial set theory
03E65 Other set-theoretic hypotheses and axioms
03E35 Consistency and independence results

Citations:

Zbl 1422.03093
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