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  1. Extremal Numberings and Fixed Point Theorems.Marat Faizrahmanov - forthcoming - Mathematical Logic Quarterly.
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  • Ordinal Analysis of Partial Combinatory Algebras.Paul Shafer & Sebastiaan A. Terwijn - 2021 - Journal of Symbolic Logic 86 (3):1154-1188.
    For every partial combinatory algebra, we define a hierarchy of extensionality relations using ordinals. We investigate the closure ordinals of pca’s, i.e., the smallest ordinals where these relations become equal. We show that the closure ordinal of Kleene’s first model is ${\omega _1^{\textit {CK}}}$ and that the closure ordinal of Kleene’s second model is $\omega _1$. We calculate the exact complexities of the extensionality relations in Kleene’s first model, showing that they exhaust the hyperarithmetical hierarchy. We also discuss embeddings of (...)
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  • Computability in Partial Combinatory Algebras.Sebastiaan A. Terwijn - 2020 - Bulletin of Symbolic Logic 26 (3-4):224-240.
    We prove a number of elementary facts about computability in partial combinatory algebras. We disprove a suggestion made by Kreisel about using Friedberg numberings to construct extensional pca’s. We then discuss separability and elements without total extensions. We relate this to Ershov’s notion of precompleteness, and we show that precomplete numberings are not 1–1 in general.
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