On constructions with 2-cardinals

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. The paper is dedicated to the memory of Jim Baumgartner whose seminal joint paper (Baumgartner and Shelah in Ann Pure Appl Logic 33(2):109–129, 1987) with Saharon Shelah provided a critical mass in the theory in question. A new result which we obtain as a side product is the consistency of the existence of a function \(f:[\lambda ^{++}]^2\rightarrow [\lambda ^{++}]^{\le \lambda }\) with the appropriate \(\lambda ^+\)-version of property \(\Delta \) for regular \(\lambda \ge \omega \) satisfying \(\lambda ^{<\lambda }=\lambda \).