Creature forcing and large continuum: The joy of halving

Abstract

For $f,g\in {\omega }^{\omega }$ let $c_{f,g}^{\forall }$ be the minimal number of uniform $g$-splitting trees needed to cover the uniform $f$-splitting tree, i.e., for every branch $\nu$ of the $f$-tree, one of the $g$-trees contains $\nu$. Let $c_{f,g}^{\exists }$ be the dual notion: For every branch $\nu$, one of the $g$-trees guesses $\nu (m)$ infinitely often. We show that it is consistent that $c_{f_{\epsilon },g_{\epsilon }}^{\exists }=c_{f_{\epsilon },g_{\epsilon }}^{\forall }={\kappa }_{\epsilon }$ for continuum many pairwise different cardinals ${\kappa }_{\epsilon }$ and suitable pairs $(f_{\epsilon },g_{\epsilon })$. For the proof we introduce a new mixed-limit creature forcing construction.

References in corpus (3)

• Decisive creatures and large continuum
• Simple Cardinal Characteristics of the Continuum
• Even more simple cardinal invariants

Cited by in corpus (8)

• Matrix iterations with vertical support restrictions
• Creature forcing and five cardinal characteristics in Cichoń's diagram
• On cardinal characteristics of Yorioka ideals
• Many Different Uniformity Numbers of Yorioka Ideals
• Preservation of splitting families and cardinal characteristics of the continuum
• Partition theorems from creatures and idempotent ultrafilters
• Continuum Many Different Things: Localisation, Anti-Localisation and Yorioka Ideals
• Some models produced by 3D iterations