Parametrized $\diamondsuit $ principles

Abstract

Abstract. We will present a collection of guessing principles which have a similar relationship to ♦ as cardinal invariants of the continuum have to CH. The purpose is to provide a means for systematically analyzing ♦ and its consequences. It also provides for a unified approach for understanding the status of a number of consequences of CH and ♦ in models such as those of Laver, Miller, and Sacks.

Gruff ultrafilters

“…We will start by laying down some terminology and results from [9] that we will be using throughout this section. The theory of parametrized diamond principles involves cardinal invariants given by triples (A, B, E) where |A| ≤ c, |B| ≤ c and E ⊆ A × B, where we additionally require that (∀a ∈ A)(∃b ∈ B)(a E b) and that (∀b ∈ B)(∃a ∈ A)¬(a E b) (these last two requirements are only to ensure existence and nontriviality of the corresponding cardinal invariant).…”

“…In order for the previous theorem to be of any use, we need to exhibit models where ♦(r P ) holds. Recall that by [9,Thm. 6.6], in many of the models of Set Theory that are obtained via countable support iterations of proper forcing notions, we will have that ♦(r P ) holds if and only if r P = ω 1 .…”

“…In this paper, we obtain three more partial answers to van Douwen's question. The first result involves the theory of diamond principles that are parametrized by a cardinal invariant, as developed in [9]. We define a cardinal characteristic r P that relates naturally to perfect subsets of Q, and show that its corresponding parametrized diamond principle ♦(r P ) implies the existence of a gruff ultrafilter which is at the same time a Q-point.…”

Covering properties which, under weak diamond principles, constrain the extents of separable spaces

“…It is clear (cf. [18]) that Φ(ω, <) is implied by the principle ♦ and so is consistent with CH. However it is an interesting open question as to whether it is independent of CH, that is, whether ZFC is consistent with ω 1 < 2 ω < 2 ω 1 and Φ(ω, <).…”

Borel Tukey morphisms and combinatorial cardinal invariants of the continuum

“…On the other hand, there are a growing number of applications of Borel morphisms appearing in the literature. See, for instance, the body of recent work on parametrized diamond principles initiated in [MHD04], or the results in Borel equivalence relations found in [CS11].…”