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| Main Authors: | , , |
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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2605.21194 |
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| _version_ | 1866914583187292160 |
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| author | Miller, Joseph S. Osso, Gian Marco Scott, Isabella |
| author_facet | Miller, Joseph S. Osso, Gian Marco Scott, Isabella |
| contents | Given a countable Turing ideal $\mathcal{I} \subseteq ω^ω$, we say that $x$ is a list (resp. weak list) of $\mathcal{I}$ if $\mathcal{I}=\{x^{[n]} : n \in ω\}$ (resp. if $\mathcal{I} \subseteq \{x^{[n]} :n \in ω\}$). 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 $Σ^0_2(x)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_21194 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Listing the hyperarithmetical functions Miller, Joseph S. Osso, Gian Marco Scott, Isabella Logic 03D60, 03D80 Given a countable Turing ideal $\mathcal{I} \subseteq ω^ω$, we say that $x$ is a list (resp. weak list) of $\mathcal{I}$ if $\mathcal{I}=\{x^{[n]} : n \in ω\}$ (resp. if $\mathcal{I} \subseteq \{x^{[n]} :n \in ω\}$). 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 $Σ^0_2(x)$. |
| title | Listing the hyperarithmetical functions |
| topic | Logic 03D60, 03D80 |
| url | https://arxiv.org/abs/2605.21194 |