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 :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 \ with the appropriate \-version of property \ for regular \ satisfying \. (shrink)

We define , a square principle in the context of , and prove its consistency relative to ZFC by a directed-closed forcing and hence that it is consistent to have hold when κ is supercompact, whereas □κ is known to fail under this condition. The new principle is then extended to produce a principle with a non-reflection property. Another variation on is also considered, this one based on a family of club subsets of . Finally, a new square principle for (...) cardinals, denoted , is introduced. This principle is proved consistent with κ being supercompact. It is shown to yield a non-reflection result similar to that given by □κ. (shrink)

We show how to construct Gitik’s short extenders gap-3 forcing using a morass, and that the forcing notion is of Prikry type.

We prove that a combinatorial consequence of the negation of the PCF conjecture for intervals, involving free subsets relative to set mappings, is not implied by even the strongest known large cardinal axiom.