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  • 1.
    Espíndola, Christian
    Stockholm University, Faculty of Science, Department of Mathematics.
    A short proof of Glivenko theorems for intermediate predicate logics2013In: Archive for mathematical logic, ISSN 0933-5846, E-ISSN 1432-0665, Vol. 52, no 7-8, p. 823-826Article in journal (Refereed)
    Abstract [en]

    We give a simple proof-theoretic argument showing that Glivenko’s theorem for propositional logic and its version for predicate logic follow as an easy consequence of the deduction theorem, which also proves some Glivenko type theorems relating intermediate predicate logics between intuitionistic and classical logic. We consider two schemata, the double negation shift (DNS) and the one consisting of instances of the principle of excluded middle for sentences (REM). We prove that both schemata combined derive classical logic, while each one of them provides a strictly weaker intermediate logic, and neither of them is derivable from the other. We show that over every intermediate logic there exists a maximal intermediate logic for which Glivenko’s theorem holds. We deduce as well a characterization of DNS, as the weakest (with respect to derivability) scheme that added to REM derives classical logic.

  • 2.
    Espíndola, Christian
    Stockholm University, Faculty of Science, Department of Mathematics.
    Achieving completeness: from constructive set theory to large cardinals2016Doctoral thesis, monograph (Other academic)
    Abstract [en]

    This thesis is an exploration of several completeness phenomena, both in the constructive and the classical settings. After some introductory chapters in the first part of the thesis where we outline the background used later on, the constructive part contains a categorical formulation of several constructive completeness theorems available in the literature, but presented here in an unified framework. We develop them within a constructive reverse mathematical viewpoint, highlighting the metatheory used in each case and the strength of the corresponding completeness theorems.

    The classical part of the thesis focuses on infinitary intuitionistic propositional and predicate logic. We consider a propositional axiomatic system with a special distributivity rule that is enough to prove a completeness theorem, and we introduce weakly compact cardinals as the adequate metatheoretical assumption for this development. Finally, we return to the categorical formulation focusing this time on infinitary first-order intuitionistic logic. We propose a first-order system with a special rule, transfinite transitivity, that embodies both distributivity as well as a form of dependent choice, and study the extent to which completeness theorems can be established. We prove completeness using a weakly compact cardinal, and, like in the constructive part, we study disjunction-free fragments as well. The assumption of weak compactness is shown to be essential for the completeness theorems to hold.

  • 3.
    Espíndola, Christian
    Stockholm University, Faculty of Science, Department of Mathematics.
    Semantic Completeness of First-Order Theories in Constructive Reverse Mathematics2016In: Notre Dame Journal of Formal Logic, ISSN 0029-4527, E-ISSN 1939-0726, Vol. 57, no 2, p. 281-286Article in journal (Refereed)
    Abstract [en]

    We introduce a general notion of semantic structure for first-order theories, covering a variety of constructions such as Tarski and Kripke semantics, and prove that, over Zermelo–Fraenkel set theory (ZF), the completeness of such semantics is equivalent to the Boolean prime ideal theorem (BPI). Using a result of McCarty (2008), we conclude that the completeness of Kripke semantics is equivalent, over intuitionistic Zermelo–Fraenkel set theory (IZF), to the Law of Excluded Middle plus BPI. Along the way, we also prove the equivalence, over ZF, between BPI and the completeness theorem for Kripke semantics for both first-order and propositional theories.

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