On constructions with $2$-cardinals

TL;DR

This paper develops a new framework using $2$-cardinals, derived from simplified morasses, to unify and extend constructions in set theory and topology, with applications to Boolean algebras, Banach spaces, and forcing.

Contribution

It introduces a novel approach replacing traditional structures with $2$-cardinals, unifying various combinatorial and forcing constructions, and proves the consistency of a new combinatorial function with property $ riangle$.

Findings

Unified theory of trees, gaps, and colorings using $2$-cardinals

Reformulation of forcing notions in the new framework

Consistency of a specialized combinatorial function with property $ riangle$

Abstract

We propose developing the theory of consequences of morasses relevant in mathematical applications in the language alternative to the usual one, replacing commonly used structures by families of sets originating with Velleman's neat simplified morasses called $2$-cardinals. The theory of related trees, gaps, colorings of pairs and forcing notions is reformulated and sketched from a unifying point of view with the focus on the applicability to constructions of mathematical structures like Boolean algebras, Banach spaces or compact spaces. A new result which we obtain as a side product is the consistency of the existence of a function $f:[\lambda^{++}]^2\rightarrow[\lambda^{++}]^{\leq\lambda}$ with the appropriate $\lambda^+$-version of property $\Delta$ for regular $\lambda\geq\omega$ satisfying $\lambda^{<\lambda}=\lambda$.