Forcing and classes of $\mathsf{HYP}$-dominating functions
Noam Greenberg, Gian Marco Osso
TL;DR
The paper proves strict separations among the hyperarithmetic non-lowness classes $\mathsf{HYP}$-$\mathsf{SNE}$, $\mathsf{HYP}$-$\mathsf{SME}$, and $\mathsf{HYP}$-$\mathsf{DOM}$ by developing effective forcing notions based on $\omega$-bushy trees and bad-sets closures, applicable to any Turing ideal closed under $\le_{\mathsf{HYP}}$. It extends the computable Cichon diagram framework to the hyperarithmetic setting, showing how effective morphisms and closure operators govern the relative strength of these problems. The constructions yield $\mathcal{I}\text{-SNE} \subsetneq \mathcal{I}\text{-SME} \subsetneq \mathcal{I}\text{-DOM}$ for a broad class of hyperarithmetic ideals, with near-optimal optimality in the sense of collapses observed in GKT. The methods provide a robust toolkit for separating non-lowness classes via forcing over countable $\mathcal{I}$ and offer relative results applicable to general $\mathcal{I}$-ideals closed under $\le_{\mathsf{HYP}}$. Overall, the work links computable cardinal characteristics to effective forcing in the hyperarithmetical realm, expanding both the theory and its potential applications.
Abstract
This paper is aimed at showing separations between three subsets of $ω^ω$, namely $\mathsf{HYP}\text{-}\mathsf{SNE}$, $\mathsf{HYP}\text{-}\mathsf{SME}$, and $\mathsf{HYP}\text{-}\mathsf{DOM}$. These classes are natural computational analogues of cardinal characteristics from Cichon's diagram and are known to satisfy $\mathsf{HYP}\text{-}\mathsf{SNE} \subseteq \mathsf{HYP}\text{-}\mathsf{SME} \subseteq \mathsf{HYP}\text{-}\mathsf{DOM}$. To show that both of these inclusions are strict we introduce effectivizations of Laver and Hechler forcing, which we believe are of independent interest. Our techniques allow us to show similar results relative to any Turing ideal closed under $\leq_{\mathsf{HYP}}$.