10 found
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  1. Classifying Positive Equivalence Relations.Claudio Bernardi & Andrea Sorbi - 1983 - Journal of Symbolic Logic 48 (3):529-538.
    Given two (positive) equivalence relations ∼ 1 , ∼ 2 on the set ω of natural numbers, we say that ∼ 1 is m-reducible to ∼ 2 if there exists a total recursive function h such that for every x, y ∈ ω, we have $x \sim_1 y \operatorname{iff} hx \sim_2 hy$ . We prove that the equivalence relation induced in ω by a positive precomplete numeration is complete with respect to this reducibility (and, moreover, a "uniformity property" holds). This (...)
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  2.  23
    The Fixed-Point Theorem for Diagonalizable Algebras.Claudio Bernardi - 1975 - Studia Logica 34 (3):239 - 251.
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  3.  35
    The Uniqueness of the Fixed-Point in Every Diagonalizable Algebra.Claudio Bernardi - 1976 - Studia Logica 35 (4):335 - 343.
    It is well known that, in Peano arithmetic, there exists a formula Theor (x) which numerates the set of theorems. By Gödel's and Löb's results, we have that Theor (˹p˺) ≡ p implies p is a theorem ∼Theor (˹p˺) ≡ p implies p is provably equivalent to Theor (˹0 = 1˺). Therefore, the considered "equations" admit, up to provable equivalence, only one solution. In this paper we prove (Corollary 1) that, in general, if P (x) is an arbitrary formula built (...)
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  4.  25
    On the Relation Provable Equivalence and on Partitions in Effectively Inseparable Sets.Claudio Bernardi - 1981 - Studia Logica 40 (1):29 - 37.
    We generalize a well-knownSmullyan's result, by showing that any two sets of the kindC a = {x/ xa} andC b = {x/ xb} are effectively inseparable (if I b). Then we investigate logical and recursive consequences of this fact (see Introduction).
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  5. Translating the Hypergame Paradox: Remarks on the Set of Founded Elements of a Relation. [REVIEW]Claudio Bernardi & Giovanna D'Agostino - 1996 - Journal of Philosophical Logic 25 (5):545 - 557.
    In Zwicker (1987) the hypergame paradox is introduced and studied. In this paper we continue this investigation, comparing the hypergame argument with the diagonal one, in order to find a proof schema. In particular, in Theorems 9 and 10 we discuss the complexity of the set of founded elements in a recursively enumerable relation on the set N of natural numbers, in the framework of reduction between relations. We also find an application in the theory of diagonalizable algebras and construct (...)
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  6.  33
    A Topological Approach to Yablo's Paradox.Claudio Bernardi - 2009 - Notre Dame Journal of Formal Logic 50 (3):331-338.
    Some years ago, Yablo gave a paradox concerning an infinite sequence of sentences: if each sentence of the sequence is 'every subsequent sentence in the sequence is false', a contradiction easily follows. In this paper we suggest a formalization of Yablo's paradox in algebraic and topological terms. Our main theorem states that, under a suitable condition, any continuous function from 2N to 2N has a fixed point. This can be translated in the original framework as follows. Consider an infinite sequence (...)
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  7.  16
    Fixed Points and Unfounded Chains.Claudio Bernardi - 2001 - Annals of Pure and Applied Logic 109 (3):163-178.
    By an unfounded chain for a function f:X→X we mean a sequence nω of elements of X s.t. fxn+1=xn for every n. Unfounded chains can be regarded as a generalization of fixed points, but on the other hand are linked with concepts concerning non-well-founded situations, as ungrounded sentences and the hypergame. In this paper, among other things, we prove a lemma in general topology, we exhibit an extensional recursive function from the set of sentences of PA into itself without an (...)
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  8.  16
    Topological Duality for Diagonalizable Algebras.Claudio Bernardi & Paola D'Aquino - 1988 - Notre Dame Journal of Formal Logic 29 (3):345-364.
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  9.  6
    Theatrum Pietatis: Images, Devotion, and Lay Drama.Claudio Bernardi - 2006 - Mediaevalia 27 (1):7-22.
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  10.  16
    A Shorter Proof of a Recent Result by R. Di Paola.Claudio Bernardi - 1984 - Notre Dame Journal of Formal Logic 25 (4):390-393.
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