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  1. A common axiom set for classical and intuitionistic plane geometry.Melinda Lombard & Richard Vesley - 1998 - Annals of Pure and Applied Logic 95 (1-3):229-255.
    We describe a first order axiom set which yields the classical first order Euclidean geometry of Tarski when used with classical logic, and yields an intuitionistic Euclidean geometry when used with intuitionistic logic. The first order language has a single six place atomic predicate and no function symbols. The intuitionistic system has a computational interpretation in recursive function theory, that is, a realizability interpretation analogous to those given by Kleene for intuitionistic arithmetic and analysis. This interpretation shows the unprovability in (...)
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  • Analyzing realizability by Troelstra's methods.Joan Rand Moschovakis - 2002 - Annals of Pure and Applied Logic 114 (1-3):203-225.
    Realizabilities are powerful tools for establishing consistency and independence results for theories based on intuitionistic logic. Troelstra discovered principles ECT 0 and GC 1 which precisely characterize formal number and function realizability for intuitionistic arithmetic and analysis, respectively. Building on Troelstra's results and using his methods, we introduce the notions of Church domain and domain of continuity in order to demonstrate the optimality of “almost negativity” in ECT 0 and GC 1 ; strengthen “double negation shift” DNS 0 to DNS (...)
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  • Separating fragments of wlem, lpo, and mp.Matt Hendtlass & Robert Lubarsky - 2016 - Journal of Symbolic Logic 81 (4):1315-1343.
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