A forcing axiom deciding the generalized Souslin Hypothesis

TL;DR

This paper introduces a new forcing axiom derived from square and diamond principles, leading to the existence of super-Souslin trees and implications for the continuum function at uncountable cardinals.

Contribution

It presents a novel forcing axiom combining square and diamond principles and applies it to establish the existence of super-Souslin trees under certain set-theoretic conditions.

Findings

Existence of super-Souslin trees under the new axiom.

If λ^{++} is not Mahlo in L, then 2^λ = λ^+ implies a λ^+-complete λ^{++}-Souslin tree.

The forcing axiom links combinatorial principles to the structure of trees at uncountable cardinals.

Abstract

We derive a forcing axiom from the conjunction of square and diamond, and present a few applications, primary among them being the existence of super-Souslin trees. It follows that for every uncountable cardinal $\lambda$, if $\lambda^{++}$ is not a Mahlo cardinal in G\"odel's constructible universe, then $2^\lambda = \lambda^+$ entails the existence of a $\lambda^+$-complete $\lambda^{++}$-Souslin tree.