Decisive creatures and large continuum

Abstract

For $f,g\inω\ho$ let $\mycfa_{f,g}$ 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 $\nu$. $\myc_{f,g}$ is the dual notion: For every branch $\nu$, one of the $g$-trees guesses $\nu (m)$ infinitely often. It is consistent that $\myc_{f_ε,g_ε}=\mycfa_{f_ε,g_ε}=κ_ε$ for $\al1$ many pairwise different cardinals ${\kappa }_{\epsilon }$ and suitable pairs $(f_{\epsilon },g_{\epsilon })$. For the proof we use creatures with sufficient bigness and halving. We show that the lim-inf creature forcing satisfies fusion and pure decision. We introduce decisiveness and use it to construct a variant of the countable support iteration of such forcings, which still satisfies fusion and pure decision.

major revision

References in corpus (1)

• Even more simple cardinal invariants

Cited by in corpus (11)

• Matrix iterations with vertical support restrictions
• Creature forcing and large continuum: The joy of halving
• On cardinal characteristics of Yorioka ideals
• Creature forcing and five cardinal characteristics in Cichoń's diagram
• Many Different Uniformity Numbers of Yorioka Ideals
• Nonproper Products
• 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
• Monotone hulls for N cap M