Applications of another characterization of βN⧹N

Abstract

Steprāns provided a characterization of βN \ N in the ℵ 2 -Cohen model that is much in the spirit of Parovičenko's characterization of this space under CH. A variety of the topological results established in the Cohen model can be deduced directly from the properties of βN \ N or P(N)/fin that feature in Steprāns' result.Cohen reals. 'The Cohen model' is any model obtained from a model of the GCH by adding a substantial quantity of Cohen reals -more than ℵ 1 . In particular 'the ℵ 2 -Cohen model' is obtained by adding ℵ 2 many Cohen reals. Actually, since we are intent on proving our results using the properties of P(N)/fin only, many readers may elect to take Lemma 2.2, Theorem 2.7 and the remark made after Proposition 2.12 on faith or else consult [18] for the necessary background on Cohen forcing.The weak Freese-Nation property. A partially ordered set P is said to have the weak Freese-Nation property if there is a function F : P → [P ] ℵ0 such that whenever p q there is r ∈ F (p) ∩ F (q) with p r q.Elementary substructures. Consider two structures M and N (groups, fields, Boolean algebras, models of set theory . . . ), where M is a substructure of N . We say that M is an elementary substructure of N , and we write M ≺ N , if every equation, involving the relations and operations of the structures and constants from M , that has a solution in N has a solution in M as well.

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.…”

The power set of ω Elementary submodels and weakenings of CH

“…See, e.g., [2] for a proof. Various weakenings of CH involve the existence of N such that B = N ∩P(ω) "captures" P(ω) in one of the following senses: Definition 2.2.…”

“…Definition 2.4 is from [2]. The usual definition of wFN(P(ω)) is in terms of wFN maps from P(ω) to [P(ω)] ≤ω , but this definition was shown in [5] to be equivalent to Definition 2.3.…”

“…There are many consequences of CH which are independent of ZFC, but are still true in Cohen models-that is, models of the form V [G], where V GCH and V [G] is a forcing extension of V obtained by adding some number (possibly 0) of Cohen reals; see [1,2,5,7,8]. Roughly, these consequences fall into two classes.…”

“…One type are elementary submodel axioms, saying that for all suitably large regular λ, there are many elementary submodels N ≺ H(λ) such that |N | = ℵ 1 and N ∩ P(ω) "captures" in some way all of P(ω); these are trivial under CH, where we could take N ∩ P(ω) = P(ω). The other are homogeneity axioms, saying that given a sequence of reals, r α : α < ω 2 , there are ω 2 of them which "look alike"; again, this is trivial under CH.In this paper, we define a new axiom, SEP, of the elementary submodel type, and explore its connection with known axioms of both types.A large number of applications of such axioms may be found in [2,4,7,8]. …”