Anatomy of $\tilde{\mathbb{E}}$

Abstract

We present a detailed general framework to describe the forcing $\tilde{\mathbb{E}}$, defined by Kellner, Shelah and Tanăsie to prove the consistency with ZFC of an alternative order of Cichoń's maximum. Our presentation is close to the framework of tree-creature forcing notions from Horowitz and Shelah. We show that the posets in this class have strong FAM limits for intervals (in recent terminology, they are $σ$-FAM-linked) and, furthermore, that they also have strong ultrafilter limits for intervals.

Figures (2)

• Figure 1: Cichoń's diagram. The arrows mean $\leq$ and dotted arrows represent $\mathop{\mathrm{\mathrm{add}}}\nolimits(\mathcal{M})=\min\{\mathfrak{b},\mathop{\mathrm{\hbox{\rm cov}}}\nolimits(\mathcal{M})\}$ and $\mathop{\mathrm{\hbox{\rm cof}}}\nolimits(\mathcal{M})=\max\{\mathfrak{d},\mathop{\mathrm{\hbox{\rm non}}}\nolimits(\mathcal{M})\}$, which we call the dependent cardinals.
• Figure 2: Cichoń's diagram after a FS iteration of (non-trivial) ccc posets with length of uncountable cofinality.

Theorems & Definitions (39)

• Definition 2.1
• Example 2.2: cf. KST
• Lemma 2.3: cf. KST
• proof
• Definition 2.4
• Lemma 2.5: cf. KST
• proof
• Definition 2.6
• Definition 2.7: KST
• proof