A microscopic approach to Souslin-tree construction, Part II

Ari Meir Brodsky; Assaf Rinot

doi:10.1016/j.apal.2020.102904

A microscopic approach to Souslin-tree construction, Part II

Ari Meir Brodsky, Assaf Rinot

Department of Mathematics

Ariel University

Research output: Contribution to journal › Article › peer-review

14 Scopus citations

Abstract

In Part I of this series, we presented the microscopic approach to Souslin-tree constructions, and argued that all known ⋄-based constructions of Souslin trees with various additional properties may be rendered as applications of our approach. In this paper, we show that constructions following the same approach may be carried out even in the absence of ⋄. In particular, we obtain a new weak sufficient condition for the existence of Souslin trees at the level of a strongly inaccessible cardinal. We also present a new construction of a Souslin tree with an ascent path, thereby increasing the consistency strength of such a tree's nonexistence from a Mahlo cardinal to a weakly compact cardinal. Section 2 of this paper is targeted at newcomers with minimal background. It offers a comprehensive exposition of the subject of constructing Souslin trees and the challenges involved.

Original language	English
Article number	102904
Journal	Annals of Pure and Applied Logic
Volume	172
Issue number	5
DOIs	https://doi.org/10.1016/j.apal.2020.102904
State	Published - May 2021

Bibliographical note

Publisher Copyright:
© 2020 Elsevier B.V.

Funding

The first author was supported by the Center for Absorption in Science, Ministry of Aliyah and Integration, State of Israel. The second author was partially supported by the European Research Council (grant agreement ERC-2018-StG 802756) and by the Israel Science Foundation (grant agreement 2066/18). This paper is submitted to the proceedings of 50 Years of Set Theory in Toronto. The two authors first met at the Toronto Set Theory Seminar, when Brodsky was a Ph.D. student of Stevo Todorcevic and Rinot was a Fields-Ontario postdoctoral fellow of Ilijas Farah, Stevo Todorcevic and Bill Weiss. We thank James Cummings, Moti Gitik, Yair Hayut, Adi Jarden, Menachem Kojman, Chris Lambie-Hanson, Philipp Lücke, Matti Rubin (of blessed memory), Stevo Todorcevic, and Bill Weiss, with whom we had stimulating discussions on the subject matter of this project throughout the years. We thank the anonymous referee for an exceptionally thorough reading of this paper and for providing a long list of valuable comments. The first author was supported by the Center for Absorption in Science, Ministry of Aliyah and Integration , State of Israel. The second author was partially supported by the European Research Council (grant agreement ERC-2018-StG 802756 ) and by the Israel Science Foundation (grant agreement 2066/18 ).

Funders	Funder number
Center for Absorption in Science
Ministry of Aliyah and Integration , State of Israel
Ministry of Aliyah and Integration, State of Israel
Horizon 2020 Framework Programme	802756
European Commission
Israel Science Foundation	2066/18

Keywords

Ascent path
Parameterized proxy principle
Postprocessing function
Souslin-tree construction
Streamlined trees
xbox

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10.1016/j.apal.2020.102904

Cite this

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A microscopic approach to Souslin-tree construction, Part II

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Keywords

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