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  1.  2
    An Extension of Shelah’s Trichotomy Theorem.Shehzad Ahmed - 2019 - Archive for Mathematical Logic 58 (1-2):137-153.
    Shelah develops the theory of \\) without the assumption that \\), going so far as to get generators for every \\) under some assumptions on I. Our main theorem is that we can also generalize Shelah’s trichotomy theorem to the same setting. Using this, we present a different proof of the existence of generators for \\) which is more in line with the modern exposition. Finally, we discuss some obstacles to further generalizing the classical theory.
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  2.  1
    The Iterability Hierarchy Above $${{Mathrm{Mathsf {}}}}$$.Alessandro Andretta & Vincenzo Dimonte - 2019 - Archive for Mathematical Logic 58 (1-2):77-97.
    In this paper we introduce a new hierarchy of large cardinals between \ and \, the iterability hierarchy, and we prove that every step of it strongly implies the ones below.
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  3.  1
    A Completeness Theorem for Continuous Predicate Modal Logic.Stefano Baratella - 2019 - Archive for Mathematical Logic 58 (1-2):183-201.
    We study a modal extension of the Continuous First-Order Logic of Ben Yaacov and Pedersen :168–190, 2010). We provide a set of axioms for such an extension. Deduction rules are just Modus Ponens and Necessitation. We prove that our system is sound with respect to a Kripke semantics and, building on Ben Yaacov and Pedersen, that it satisfies a number of properties similar to those of first-order predicate logic. Then, by means of a canonical model construction, we get that every (...)
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  4.  5
    The Binary Expansion and the Intermediate Value Theorem in Constructive Reverse Mathematics.Josef Berger, Hajime Ishihara, Takayuki Kihara & Takako Nemoto - 2019 - Archive for Mathematical Logic 58 (1-2):203-217.
    We introduce the notion of a convex tree. We show that the binary expansion for real numbers in the unit interval ) is equivalent to weak König lemma ) for trees having at most two nodes at each level, and we prove that the intermediate value theorem is equivalent to \ for convex trees, in the framework of constructive reverse mathematics.
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  5.  2
    Convexity and Unique Minimum Points.Josef Berger & Gregor Svindland - 2019 - Archive for Mathematical Logic 58 (1-2):27-34.
    We show constructively that every quasi-convex, uniformly continuous function \ with at most one minimum point has a minimum point, where C is a convex compact subset of a finite dimensional normed space. Applications include a result on strictly quasi-convex functions, a supporting hyperplane theorem, and a short proof of the constructive fundamental theorem of approximation theory.
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  6.  7
    Sequent Calculus for Classical Logic Probabilized.Marija Boričić - 2019 - Archive for Mathematical Logic 58 (1-2):119-136.
    Gentzen’s approach to deductive systems, and Carnap’s and Popper’s treatment of probability in logic were two fruitful ideas that appeared in logic of the mid-twentieth century. By combining these two concepts, the notion of sentence probability, and the deduction relation formalized in the sequent calculus, we introduce the notion of ’probabilized sequent’ \ with the intended meaning that “the probability of truthfulness of \ belongs to the interval [a, b]”. This method makes it possible to define a system of derivations (...)
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  7.  2
    Diagonal Reflections on Squares.Gunter Fuchs - 2019 - Archive for Mathematical Logic 58 (1-2):1-26.
    The effects of the forcing axioms \, \ and \ on the failure of weak threaded square principles of the form \\) are analyzed. To this end, a diagonal reflection principle, \, and it implies the failure of \\) if \. It is also shown that this result is sharp. It is noted that \/\ imply the failure of \\), for every regular \, and that this result is sharp as well.
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  8.  1
    Strange Ultrafilters.Moti Gitik - 2019 - Archive for Mathematical Logic 58 (1-2):35-52.
    We deal with some natural properties of ultrafilters which trivially fail for normal ultrafilters.
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  9.  3
    A Model of the Generic Vopěnka Principle in Which the Ordinals Are Not Mahlo.Victoria Gitman & Joel David Hamkins - 2019 - Archive for Mathematical Logic 58 (1-2):245-265.
    The generic Vopěnka principle, we prove, is relatively consistent with the ordinals being non-Mahlo. Similarly, the generic Vopěnka scheme is relatively consistent with the ordinals being definably non-Mahlo. Indeed, the generic Vopěnka scheme is relatively consistent with the existence of a \-definable class containing no regular cardinals. In such a model, there can be no \-reflecting cardinals and hence also no remarkable cardinals. This latter fact answers negatively a question of Bagaria, Gitman and Schindler.
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  10.  6
    Selfextensional Logics with a Distributive Nearlattice Term.Luciano J. González - 2019 - Archive for Mathematical Logic 58 (1-2):219-243.
    We define when a ternary term m of an algebraic language \ is called a distributive nearlattice term -term) of a sentential logic \. Distributive nearlattices are ternary algebras generalising Tarski algebras and distributive lattices. We characterise the selfextensional logics with a \-term through the interpretation of the DN-term in the algebras of the algebraic counterpart of the logics. We prove that the canonical class of algebras associated with a selfextensional logic with a \-term is a variety, and we obtain (...)
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  11.  2
    Uniform Interpolation and Sequent Calculi in Modal Logic.Rosalie Iemhoff - 2019 - Archive for Mathematical Logic 58 (1-2):155-181.
    A method is presented that connects the existence of uniform interpolants to the existence of certain sequent calculi. This method is applied to several modal logics and is shown to cover known results from the literature, such as the existence of uniform interpolants for the modal logic \. New is the result that \ has uniform interpolation. The results imply that for modal logics \ and \, which are known not to have uniform interpolation, certain sequent calculi cannot exist.
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  12.  1
    A Strong Failure of $$Aleph _0$$-Stability for Atomic Classes.Michael C. Laskowski & Saharon Shelah - 2019 - Archive for Mathematical Logic 58 (1-2):99-118.
    We study classes of atomic models \ of a countable, complete first-order theory T. We prove that if \ is not \-small, i.e., there is an atomic model N that realizes uncountably many types over \\) for some finite \ from N, then there are \ non-isomorphic atomic models of T, each of size \.
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  13.  2
    Families of Sets Related to Rosenthal’s Lemma.Damian Sobota - 2019 - Archive for Mathematical Logic 58 (1-2):53-69.
    A family \ is called Rosenthal if for every Boolean algebra \, bounded sequence \ of measures on \, antichain \ in \, and \, there exists \ such that \<\varepsilon \) for every \. Well-known and important Rosenthal’s lemma states that \ is a Rosenthal family. In this paper we provide a necessary condition in terms of antichains in \}\) for a family to be Rosenthal which leads us to a conclusion that no Rosenthal family has cardinality strictly less (...)
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  14.  2
    Extendible Cardinals and the Mantle.Toshimichi Usuba - 2019 - Archive for Mathematical Logic 58 (1-2):71-75.
    The mantle is the intersection of all ground models of V. We show that if there exists an extendible cardinal then the mantle is the smallest ground model of V.
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