A Microscopic approach to Souslin-tree constructions. Part I

TL;DR

This paper introduces a new parameterized proxy principle and a microscopic method for constructing $ ext{Souslin}$ trees with various features, applicable across different cardinals including inaccessible ones, unifying and extending previous constructions.

Contribution

It presents a novel proxy principle and a microscopic approach for constructing $ ext{Souslin}$ trees, broadening applicability and unifying existing $ ext{diamondsuit}$-based methods.

Findings

Constructed a coherent $ ext{Souslin}$ tree for inaccessible cardinals.

Established the consistency of various proxy principle instances.

Unified previous $ ext{diamondsuit}$-based $ ext{Souslin}$ tree constructions.

Abstract

We propose a parameterized proxy principle from which $\kappa$-Souslin trees with various additional features can be constructed, regardless of the identity of $\kappa$. We then introduce the microscopic approach, which is a simple method for deriving trees from instances of the proxy principle. As a demonstration, we give a construction of a coherent $\kappa$-Souslin tree that applies also for $\kappa$ inaccessible. We then carry out a systematic study of the consistency of instances of the proxy principle, distinguished by the vector of parameters serving as its input. Among other things, it will be shown that all known $\diamondsuit$-based constructions of $\kappa$-Souslin trees may be redirected through this new proxy principle.