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 [5] 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 : [λ ++ ] 2 → [λ ++ ] ≤λ with the appropriate λ + -version of property ∆ for regular λ ≥ ω satisfying λ <λ = λ.

On strong chains of sets and functions

“…In [34] and [32] one may find more open problems. Two other papers on topics related to this paper which we have not mentioned are [17] and [15]. In the former, Laver shows, among other things, why one cannot approach the question of Hajnal and Szentmiklóssy in the usual way as in the Baire space, which is to say, by iteratively adding functions dominating modulo finite.…”

On $\mathbb R$-embeddability of almost disjoint families and Akemann–Doner C$^*$-algebras

“…where for two sets A, B of ordinals A < B means that α < β for any α ∈ A and β ∈ B. For more details on the structure of µ see [19…”

“…and α ∈ Y = Y 1 * Y 2 and the ranks of X 1 , X 2 , Y 1 , Y 2 are elements of µ of fixed rank n ∈ N. By [19, Definition 1.1(3)] there is an order preserving f Y,X : X → Y , which by [19, Definition 1.1 (3,5)] must satisfy f [X 1 ] = Y 1 and f [X 2 ] = Y 2 and moreover f (X ∩ (α + 1)) is the identity on X ∩ (α + 1) be the coherence lemma 2.1 of [19], so f Y,X (α) = α and f [X…”

“…follows from the definition of c, i.e., from the minimality of the rank of X α, β, which is of the form To obtain (5), recall [19,Theorem 4.5…”

Characterizations of the weakly compact ideal on Pλ

“…Koszmider (see [Kos17,Proposition 4.3] or [Kos95, Proposition 10]) has shown that if µ is a (κ, λ)-semimorass then it satisfies the following non-reflection property: for every proper subset X λ with |X| ≥ κ we have µ ∩ P κ X ∈ NS κ,X .…”