Large cardinals and gap-1 morasses

TL;DR

This paper introduces a new forcing method to create morasses at all regular uncountable cardinals, preserving large cardinals, and explores the existence of universal morasses within this framework.

Contribution

It develops a novel forcing technique with homogeneity properties to produce morasses at every regular uncountable cardinal while maintaining large cardinal axioms.

Findings

Existence of a new forcing with homogeneity properties called mangroves.

Morasses can be constructed at all regular uncountable cardinals.

Universal morasses are consistent with strong large cardinal axioms.

Abstract

We present a new partial order for directly forcing morasses to exist that enjoys a significant homogeneity property. We then use this forcing in a reverse Easton iteration to obtain an extension universe with morasses at every regular uncountable cardinal, while preserving all n-superstrong (0<n<omega+1), hyperstrong and 1-extendible cardinals. In the latter case, a preliminary forcing to make the GCH hold is required. Our forcing yields morasses that satisfy an extra property related to the homogeneity of the partial order; we refer to them as mangroves and prove that their existence is equivalent to the existence of morasses. Finally, we exhibit a partial order that forces universal morasses to exist at every regular uncountable cardinal, and use this to show that universal morasses are consistent with n-superstrong, hyperstrong, and 1-extendible cardinals. This all contributes to the second author's outer model programme, the aim of which is to show that L-like principles can hold in outer models which nevertheless contain large cardinals.