On the weak Freese–Nation property of ?(ω)

Abstract

Continuing [6], [8] and [16], we study the consequences of the weak FreeseNation property of (P(ω), ⊆). Under this assumption, we prove that most of the known cardinal invariants including all of those appearing in Cichoń's diagram take the same value as in the corresponding Cohen model. Using this principle we could also strengthen two results of W. Just about cardinal sequences of superatomic Boolean algebras in a Cohen model. These results show that the weak Freese-Nation property of (P(ω), ⊆) captures many of the features of Cohen models and hence may be considered as a principle axiomatizing a good portion of the combinatorics available in Cohen models.

Some combinatorial principles defined in terms of elementary submodels

“…Our formulation shows that many of the consequences of the weak Freese-Nation property of P(ω) studied in [6] already follow from SEP. We show that it is consistent that SEP holds while P(ω) fails to have the (ℵ 1 , ℵ 0 )-ideal property introduced in [2]. This answers a question addressed independently by Fuchino and by Kunen.…”

“…A It was also proved in [6] that WFN(P(ω)) implies that a, the smallest size of a maximal almost disjoint family in P(ω), is ℵ 1 . In the proof it is sufficient to assume IDP(P(ω)).…”

“…In [6] many interesting consequences of WFN(P(ω)) have been found. Concerning the combinatorics of the reals, a universe satisfying WFN(P(ω)) behaves very similarly to a Cohen model.…”

“…In the proofs of most of the results in [6] it is only used that under WFN(P(ω)), for some sufficiently large χ there are cofinally many M ∈ M χ with P(ω)∩M ≤ σ P(ω). The following theorem collects some of the consequences of SEP(P(ω)) that follow from the arguments given in [6].…”

Coloring ordinals by reals

“…9 Finally, we shall mention some open problems. In [6] it is shown that a = ℵ 1 follows from WFN where a is the almost disjoint number. In [4], it is then shown that, under some additional assumptions, a = ℵ 1 already follows from SEP which is a weakening of WFN.…”