Microscopic approach to Souslin-tree construction

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How to construct a Souslin tree the right way Here we demonstrate a simple construction of a κ-Souslin tree, for arbitrary regular uncountable cardinal κ, from a straightforward combinatorial axiom.


A Microscopic approach to Souslin-tree constructions. Part I  (Ari Meir Brodsky and Assaf Rinot, Annals of Pure and Applied Logic, to appear, 2017)
Here we present the goals of the project and introduce the proxy principle as well as the microscopic approach.  We analyze the properties of the simplest Souslin tree constructed from the proxy principle, and we show how to construct Souslin trees with additional properties, such as completeness and uniform coherence.  We explain the differences between the classical approach to building Souslin trees and our new approach.  The bulk of the paper consists of building the bridge between the old and new foundations.  That is, we show how all known diamond-based constructions of Souslin trees may be redirected through the proxy principle.

Reduced powers of Souslin trees  (Ari Meir Brodsky and Assaf Rinot, Forum of Mathematics, Sigma, 5(e2), pp. 1-82, 2017)
Here we use the microscopic approach to build κ-Souslin trees with sophisticated properties, including various prescribed combinations of freeness degree and ascent-path characteristics.  As a sample corollary, we obtain the consistency of an ℵ6-Souslin tree T and a sequence of uniform ultrafilters 〈𝒰n : n < 6〉 such that Tn/𝒰n is ℵ6-Aronszajn if and only if n < 6 is not a prime number.

More notions of forcing add a Souslin tree  (Ari Meir Brodsky and Assaf Rinot, arXiv:1607.07033, Submitted July 2016)
Here we show that by taking extra care in the construction, we can obtain a Souslin tree from a weaker version of the proxy principle than shown in the previous papers.  We then identify a large class of notions of forcing that, assuming a GCH-type assumption, add a 𝜆+-Souslin tree.  This class includes Prikry, Magidor and Radin forcing.

Higher Souslin trees and the GCH, revisited  (Assaf Rinot, Advances in Mathematics, 311(C):510-531, 2017)
Here we prove that for every uncountable cardinal 𝜆, GCH + ⃞(𝜆+) entails an instance of the proxy principle at 𝜆+, implying, in particular, that if GCH holds and there are no ℵ2-Souslin trees, then ℵ2 is a weakly compact cardinal in 𝐿, solving a problem that was open for 40 years.

Distributive Aronszajn trees  (Ari Meir Brodsky and Assaf Rinot, Submitted April 2017)
A major component of this paper is the study of postprocessing functions and their effect on square sequences.  This will enable a cleaner exposition of the bridge between the old and new foundations, making it easier to identify scenarios in which Souslin trees can be shown to exist.  In particular, ⃞(𝜆+)  is sufficient to construct a uniformly coherent 𝜆+-Souslin tree, assuming that 𝜆 is a strong-limit singular cardinal and 2𝜆 = 𝜆+.

Square with built-in diamond-plus  (Assaf Rinot and Ralf Schindler, Journal of Symbolic Logic, to appear, 2017)

Reflection on the coloring and chromatic numbers  (Chris Lambie-Hanson and Assaf Rinot, Submitted December 2016)

Aronszajn trees, square principles, and stationary reflection  (Chris Lambie-Hanson, Mathematical Logic Quarterly, to appear, 2017)

Slide presentations

The current state of the Souslin problem (Assaf Rinot), ASL North American Annual Meeting, Boise, Idaho, March 2017

Coloring vs. Chromatic (Assaf Rinot), MFO workshop in Set Theory, Obwerwolfach, February 2017

The ℵ2-Souslin problem (Assaf Rinot), Set Theory and its Applications in Topology, Oaxaca, September 2016

Custom-made Souslin trees (Ari Meir Brodsky),  Logic, Set Theory and Topology seminar, Ben-Gurion University of the Negev, Beer Sheva, May 2016

Custom-made Souslin trees (Ari Meir Brodsky),  Toronto Set Theory Seminar, Fields Institute, Toronto, May 2016

Custom-made Souslin trees (Ari Meir Brodsky), Kurt Gödel Research Center, Vienna, January 2016

A new framework for Souslin-tree constructions (Ari Meir Brodsky),  The 19th Midrasha Mathematicae, 8th Young Set Theory Workshop (Compactness, Incompactness and Canonical Structures), Israel Institute for Advanced Studies, Jerusalem, October 2015 (includes video recording)

3-trees (Assaf Rinot),  P.O.I. Workshop in pure and descriptive set theory, Torino, September 2015

A microscopic approach to higher Souslin-tree constructions (Ari Meir Brodsky), Boise Extravaganza in Set Theory, San Francisco, June 2015

A microscopic approach to Souslin trees constructions (Assaf Rinot),  Forcing and its Applications Retrospective Workshop, Toronto, April 2015