Gregory Trees, The Continuum, And Martin's Axiom

TL;DR

This paper explores the existence of Gregory trees within models of Martin's Axiom where the continuum size varies, advancing understanding of their relationship with set-theoretic properties.

Contribution

It constructs models of MA with arbitrarily large continuum that either contain or lack Gregory trees, extending previous investigations into their set-theoretic behavior.

Findings

Models of MA with large continuum can have Gregory trees.

Models of MA with large continuum can lack Gregory trees.

The existence of Gregory trees is independent of the continuum size under MA.

Abstract

We continue the investigation of Gregory trees and the Cantor Tree Property carried out by Hart and Kunen. We produce models of MA with the Continuum arbitrarily large in which there are Gregory trees, and in which there are no Gregory trees.