6 found
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  1.  3
    Completeness of the primitive recursive $$omega $$-rule.Emanuele Frittaion - forthcoming - Archive for Mathematical Logic:1-17.
    Shoenfield’s completeness theorem states that every true first order arithmetical sentence has a recursive \-proof encodable by using recursive applications of the \-rule. For a suitable encoding of Gentzen style \-proofs, we show that Shoenfield’s completeness theorem applies to cut free \-proofs encodable by using primitive recursive applications of the \-rule. We also show that the set of codes of \-proofs, whether it is based on recursive or primitive recursive applications of the \-rule, is \ complete. The same \ completeness (...)
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  2.  11
    Linear Extensions of Partial Orders and Reverse Mathematics.Emanuele Frittaion & Alberto Marcone - 2012 - Mathematical Logic Quarterly 58 (6):417-423.
    We introduce the notion of τ-like partial order, where τ is one of the linear order types ω, ω*, ω + ω*, and ζ. For example, being ω-like means that every element has finitely many predecessors, while being ζ-like means that every interval is finite. We consider statements of the form “any τ-like partial order has a τ-like linear extension” and “any τ-like partial order is embeddable into τ” . Working in the framework of reverse mathematics, we show that these (...)
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  3.  8
    Reverse Mathematics and Initial Intervals.Emanuele Frittaion & Alberto Marcone - 2014 - Annals of Pure and Applied Logic 165 (3):858-879.
    In this paper we study the reverse mathematics of two theorems by Bonnet about partial orders. These results concern the structure and cardinality of the collection of initial intervals. The first theorem states that a partial order has no infinite antichains if and only if its initial intervals are finite unions of ideals. The second one asserts that a countable partial order is scattered and does not contain infinite antichains if and only if it has countably many initial intervals. We (...)
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  4.  6
    On Goodman Realizability.Emanuele Frittaion - 2019 - Notre Dame Journal of Formal Logic 60 (3):523-550.
    Goodman’s theorem states that HAω+AC+RDC is conservative over HA. The same result applies to the extensional case, that is, E-HAω+AC+RDC is also conservative over HA. This is due to Beeson. In this article, we modified the Goodman realizability and provide a new proof of the extensional case.
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  5.  13
    Reverse Mathematics, Well-Quasi-Orders, and Noetherian Spaces.Emanuele Frittaion, Matthew Hendtlass, Alberto Marcone, Paul Shafer & Jeroen Van der Meeren - 2016 - Archive for Mathematical Logic 55 (3-4):431-459.
    A quasi-order Q induces two natural quasi-orders on P\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{P}}$$\end{document}, but if Q is a well-quasi-order, then these quasi-orders need not necessarily be well-quasi-orders. Nevertheless, Goubault-Larrecq, pp. 453–462, 2007) showed that moving from a well-quasi-order Q to the quasi-orders on P\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{P}}$$\end{document} preserves well-quasi-orderedness in a topological sense. Specifically, Goubault-Larrecq proved that the upper topologies of the induced quasi-orders on P\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} (...)
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  6.  1
    Reverse Mathematics and Partial Orders, University of Udine, Italy, 2014. Supervised by Alberto Marcone.Emanuele Frittaion - 2018 - Bulletin of Symbolic Logic 24 (2):196-196.
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