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arXiv · 2606.24640

Higher Type ITTM-recursion and Determinacy of Infinite Games

Philip D Welch
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Abstract

We outline a theory of type 2 recursion for Infinite Time Turing Machines {\em \`a la Kleene}. We establish a connection between classical descriptive set theory and ittm theory, by calculating the complexity of its halting problem as exactly that of a complete $\Game \Sigma^0_3$ (or $G_{\delta\sigma}$) set. This mirrors exactly what Kleene, Moschovakis {\em et al.} achieved for Kleene's type 2 recursion and $\Sigma^0_1$ (or Open) Determinacy.} We ascertain the least ordinal which is not generalised recursive in this sense, and its characterisation {\via}a concept of {\em infinite nestings} in G\"odel's constructible hierarchy. The results do not require large cardinal axioms, and are all provable within analysis.

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Philip D Welch. 2026-06-23. Higher Type ITTM-recursion and Determinacy of Infinite Games. https://arxiv.org/abs/2606.24640

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