Creature forcing and large continuum: The joy of halving

TL;DR

This paper introduces a new mixed-limit creature forcing method to demonstrate the consistency of continuum many distinct large cardinal invariants related to splitting trees, highlighting the richness of the continuum's structure.

Contribution

It develops a novel mixed-limit creature forcing technique to establish the consistency of continuum many distinct splitting tree invariants.

Findings

Consistency of continuum many different large cardinal invariants

Introduction of a new mixed-limit creature forcing construction

Demonstration of the rich structure of the continuum

Abstract

For $f,g\in\omega^\omega$ let $c^\forall_{f,g}$ 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 $\nu$. Let $c^\exists_{f,g}$ 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 $c^\exists_{f_\epsilon,g_\epsilon}=c^\forall_{f_\epsilon,g_\epsilon}=\kappa_\epsilon$ 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.