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 $ν$ of the $f$-tree, one of the $g$-trees contains $ν$. $\myc_{f,g}$ is the dual notion: For every branch $ν$, one of the $g$-trees guesses $ν(m)$ infinitely often. It is consistent that $\myc_{f_ε,g_ε}=\mycfa_{f_ε,g_ε}=κ_ε$ for $\al1$ many pairwise different cardinals $κ_ε$ and suitable pairs $(f_ε,g_ε)$. 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