Listing the hyperarithmetical functions

TL;DR

This paper investigates the relationship between hyperarithmetical functions and lists of countable Turing ideals, providing characterizations and constructing examples that resolve open problems in the field.

Contribution

It establishes equivalences between computing lists of ideals and dominating functions, and constructs reals with specific properties that answer open questions.

Findings

For certain ideals, x computes a list iff it dominates all functions in the ideal.

Constructs reals that are HYP-strongly null engulfing but cannot compute a weak list for HYP.

Characterizes reals that compute a list of HYP as exactly those that are HYP-dominating and have  as _2(x).

Abstract

Given a countable Turing ideal $\mathcal{I} \subseteq \omega^{\omega}$, we say that $x$ is a list (resp. weak list) of $\mathcal{I}$ if $\mathcal{I}=\{x^{[n]} : n \in \omega\}$ (resp. if $\mathcal{I} \subseteq \{x^{[n]} :n \in \omega\}$). We show that, for several natural ideals $\mathcal{I}$, $x$ computes a list of $\mathcal{I}$ if and only if it computes a function dominating all the functions in $\mathcal{I}$. On the other hand, we provide reals which are $\mathsf{HYP}$-strongly null engulfing (and hence $\mathsf{HYP}$-dominating, by results of Greenberg, Kuyper and Turetsky) but which cannot compute a weak list for $\mathsf{HYP}$, solving a problem left open in a recent paper by Greenberg and the second author. This result can be generalized to any countable ideal which is downward closed under $\leq_{\mathsf{HYP}}$. We also give a characterization of reals which compute a list of $\mathsf{HYP}$: $x$ computes a list of $\mathsf{HYP}$ if and only if $x$ is $\mathsf{HYP}$-dominating and $\mathcal{O}$ is $\Sigma^0_2(x)$.