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  1. Extension, Translation, and the Cantor-Bernstein Property.Thomas William Barrett & Hans Halvorson - manuscript
    The purpose of this paper is to examine in detail a particularly interesting pair of first-order theories. In addition to clarifying the overall geography of notions of equivalence between theories, this simple example yields two surprising conclusions about the relationships that theories might bear to one another. In brief, we see that theories lack both the Cantor-Bernstein and co-Cantor-Bernstein properties.
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  2. On Representations of Intended Structures in Foundational Theories.Neil Barton, Moritz Müller & Mihai Prunescu - manuscript
    Often philosophers, logicians, and mathematicians employ a notion of intended structure when talking about a branch of mathematics. In addition, we know that there are foundational mathematical theories that can find representatives for the objects of informal mathematics. In this paper, we examine how faithfully foundational theories can represent intended structures, and show that this question is closely linked to the decidability of the theory of the intended structure. We argue that this sheds light on the trade-off between expressive power (...)
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  3. Logics of Formal Inconsistency Enriched with Replacement: An Algebraic and Modal Account.Walter Carnielli, Marcelo E. Coniglio & David Fuenmayor - manuscript
    One of the most expected properties of a logical system is that it can be algebraizable, in the sense that an algebraic counterpart of the deductive machinery could be found. Since the inception of da Costa's paraconsistent calculi, an algebraic equivalent for such systems have been searched. It is known that these systems are non self-extensional (i.e., they do not satisfy the replacement property). More than this, they are not algebraizable in the sense of Blok-Pigozzi. The same negative results hold (...)
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  4. A BIBLIOGRAPHY: JOHN CORCORAN's PUBLICATIONS ON ARISTOTLE 1972–2015.John Corcoran - manuscript
    This presentation includes a complete bibliography of John Corcoran’s publications devoted at least in part to Aristotle’s logic. Sections I–IV list 20 articles, 43 abstracts, 3 books, and 10 reviews. It starts with two watershed articles published in 1972: the Philosophy & Phenomenological Research article that antedates Corcoran’s Aristotle’s studies and the Journal of Symbolic Logic article first reporting his original results; it ends with works published in 2015. A few of the items are annotated with endnotes connecting them with (...)
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  5. Computational Reverse Mathematics and Foundational Analysis.Benedict Eastaugh - manuscript
    Reverse mathematics studies which subsystems of second order arithmetic are equivalent to key theorems of ordinary, non-set-theoretic mathematics. The main philosophical application of reverse mathematics proposed thus far is foundational analysis, which explores the limits of different foundations for mathematics in a formally precise manner. This paper gives a detailed account of the motivations and methodology of foundational analysis, which have heretofore been largely left implicit in the practice. It then shows how this account can be fruitfully applied in the (...)
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  6. Partitions and Objective Indefiniteness.David Ellerman - manuscript
    Classical physics and quantum physics suggest two meta-physical types of reality: the classical notion of a objectively definite reality with properties "all the way down," and the quantum notion of an objectively indefinite type of reality. The problem of interpreting quantum mechanics (QM) is essentially the problem of making sense out of an objectively indefinite reality. These two types of reality can be respectively associated with the two mathematical concepts of subsets and quotient sets (or partitions) which are category-theoretically dual (...)
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  7. Contradiction From a Coded Proof Predicate (Very Tentative Draft, Do Not Cite).T. Parent - manuscript
  8. Paradox with Just Self-Reference.T. Parent - manuscript
    If a semantically open language has no constraints on self-reference, one can prove an absurdity. The argument utilizes co-referring names 'a0' and 'a1', and the definition of a functional expression 'The reflection of x = y'. The definition enables a type of self-reference without deploying any semantic terminology--yet given that a0= a1, the definition implies the that 'a0' = 'a1', which is absurd. In truth, however, 'the reflection of x = y' expresses an ill-defined function. And since there is a (...)
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  9. Contradiction From the Fixed Point Lemma.T. Parent - manuscript
    Assuming that PRA (primitive recursive arithmetic) is consistent and Church's Thesis is true, the Fixed Point Lemma enables the derivation of a contradiction, in a manner akin to the v-Curry paradox. The derivation could be blocked by excluding certain formulae from the scope of the Lemma, but this would effectively concede that the unrestricted Lemma is false. The resolution of the paradox thus remains an open question.
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  10. Category Theory: A Gentle Introduction.Peter Smith - manuscript
    This Gentle Introduction is very much still work in progress. Roughly aimed at those who want something a bit more discursive, slower-moving, than Awodey's or Leinster's excellent books. -/- The current [Jan 2018] version is 291pp.
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  11. Informal and Formal Proofs, Metalogic, and the Groundedness Problem.Mario Bacelar Valente - manuscript
    When modeling informal proofs like that of Euclid’s Elements using a sound logical system, we go from proofs seen as somewhat unrigorous – even having gaps to be filled – to rigorous proofs. However, metalogic grounds the soundness of our logical system, and proofs in metalogic are not like formal proofs and look suspiciously like the informal proofs. This brings about what I am calling here the groundedness problem: how can we decide with certainty that our metalogical proofs are rigorous (...)
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  12. Metrically Homogeneous Graphs of Diameter Three.Daniela Amato, Gregory Cherlin & H. Dugald Macpherson - forthcoming - Journal of Mathematical Logic.
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  13. Dependent Choice, Properness, and Generic Absoluteness.David Asperó & Asaf Karagila - forthcoming - Review of Symbolic Logic:1-25.
    We show that Dependent Choice is a sufficient choice principle for developing the basic theory of proper forcing, and for deriving generic absoluteness for the Chang model in the presence of large cardinals, even with respect to $\mathsf {DC}$ -preserving symmetric submodels of forcing extensions. Hence, $\mathsf {ZF}+\mathsf {DC}$ not only provides the right framework for developing classical analysis, but is also the right base theory over which to safeguard truth in analysis from the independence phenomenon in the presence of (...)
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  14. How Much Propositional Logic Suffices for Rosser’s Essential Undecidability Theorem?Guillermo Badia, Petr Cintula, Petr Hajek & Andrew Tedder - forthcoming - Review of Symbolic Logic:1-18.
    In this paper we explore the following question: how weak can a logic be for Rosser's essential undecidability result to be provable for a weak arithmetical theory? It is well known that Robinson's Q is essentially undecidable in intuitionistic logic, and P. Hajek proved it in the fuzzy logic BL for Grzegorczyk's variant of Q which interprets the arithmetic operations as non-total non-functional relations. We present a proof of essential undecidability in a much weaker substructural logic and for a much (...)
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  15. Structural Relativity and Informal Rigour.Neil Barton - forthcoming - In Objects, Structures, and Logics, FilMat Studies in the Philosophy of Mathematics.
    Informal rigour is the process by which we come to understand particular mathematical structures and then manifest this rigour through axiomatisations. Structural relativity is the idea that the kinds of structures we isolate are dependent upon the logic we employ. We bring together these ideas by considering the level of informal rigour exhibited by our set-theoretic discourse, and argue that different foundational programmes should countenance different underlying logics (intermediate between first- and second-order) for formulating set theory. By bringing considerations of (...)
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  16. The Well-Ordered Society Under Crisis: A Formal Analysis of Public Reason Vs. Convergence Discourse.Hun Chung - forthcoming - American Journal of Political Science:1-20.
    A well-ordered society faces a crisis whenever a sufficient number of noncompliers enter into the political system. This has the potential to destabilize liberal democratic political order. This article provides a formal analysis of two competing solutions to the problem of political stability offered in the public reason liberalism literature—namely, using public reason or using convergence discourse to restore liberal democratic political order in the well-ordered society. The formal analyses offered in this article show that using public reason fails completely, (...)
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  17. Definability Aspects of the Denjoy Integral.Walsh Sean - forthcoming - Fundamenta Mathematicae.
    The Denjoy integral is an integral that extends the Lebesgue integral and can integrate any derivative. In this paper, it is shown that the graph of the indefinite Denjoy integral f↦∫xaf is a coanalytic non-Borel relation on the product space M[a,b]×C[a,b], where M[a,b] is the Polish space of real-valued measurable functions on [a,b] and where C[a,b] is the Polish space of real-valued continuous functions on [a,b]. Using the same methods, it is also shown that the class of indefinite Denjoy integrals, (...)
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  18. Natural Formalization: Deriving the Cantor-Bernstein Theorem in Zf.Wilfried Sieg & Patrick Walsh - forthcoming - Review of Symbolic Logic:1-44.
    Natural Formalization proposes a concrete way of expanding proof theory from the meta-mathematical investigation of formal theories to an examination of “the concept of the specifically mathematical proof.” Formal proofs play a role for this examination in as much as they reflect the essential structure and systematic construction of mathematical proofs. We emphasize three crucial features of our formal inference mechanism: (1) the underlying logical calculus is built for reasoning with gaps and for providing strategic directions, (2) the mathematical frame (...)
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  19. A Note on Choice Principles in Second-Order Logic.Benjamin Siskind, Paolo Mancosu & Stewart Shapiro - forthcoming - Review of Symbolic Logic:1-12.
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  20. Notice of Retraction: Pseudofinite Difference Field.Tingxiang Zou - forthcoming - Journal of Mathematical Logic:1993001.
    Journal of Mathematical Logic, Ahead of Print.
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  21. Double Negation Semantics for Generalisations of Heyting Algebras.Rob Arthan & Paulo Oliva - 2021 - Studia Logica 109 (2):341-365.
    This paper presents an algebraic framework for investigating proposed translations of classical logic into intuitionistic logic, such as the four negative translations introduced by Kolmogorov, Gödel, Gentzen and Glivenko. We view these as variant semantics and present a semantic formulation of Troelstra’s syntactic criteria for a satisfactory negative translation. We consider how each of the above-mentioned translation schemes behaves on two generalisations of Heyting algebras: bounded pocrims and bounded hoops. When a translation fails for a particular class of algebras, we (...)
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  22. Algebraic Combinatorics in Bounded Induction.Joaquín Borrego-Díaz - 2021 - Annals of Pure and Applied Logic 172 (2):102885.
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  23. Filter-Linkedness and its Effect on Preservation of Cardinal Characteristics.Jörg Brendle, Miguel A. Cardona & Diego A. Mejía - 2021 - Annals of Pure and Applied Logic 172 (1):102856.
    We introduce the property “F-linked” of subsets of posets for a given free filter F on the natural numbers, and define the properties “μ-F-linked” and “θ-F-Knaster” for posets in a natural way. We show that θ-F-Knaster posets preserve strong types of unbounded families and of maximal almost disjoint families. Concerning iterations of such posets, we develop a general technique to construct θ-Fr-Knaster posets (where Fr is the Frechet ideal) via matrix iterations of <θ-ultrafilter-linked posets (restricted to some level of the (...)
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  24. On the Correspondence Between Nested Calculi and Semantic Systems for Intuitionistic Logics.Tim Lyon - 2021 - Journal of Logic and Computation 31 (1):213-265.
    This paper studies the relationship between labelled and nested calculi for propositional intuitionistic logic, first-order intuitionistic logic with non-constant domains and first-order intuitionistic logic with constant domains. It is shown that Fitting’s nested calculi naturally arise from their corresponding labelled calculi—for each of the aforementioned logics—via the elimination of structural rules in labelled derivations. The translational correspondence between the two types of systems is leveraged to show that the nested calculi inherit proof-theoretic properties from their associated labelled calculi, such as (...)
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  25. |˜-Divisibility of Ultrafilters.Boris Šobot - 2021 - Annals of Pure and Applied Logic 172 (1):102857.
    We further investigate a divisibility relation on the set of BN ultrafilters on the set of natural numbers. We single out prime ultrafilters (divisible only by 1 and themselves) and establish a hierarchy in which a position of every ultrafilter depends on the set of prime ultrafilters it is divisible by. We also construct ultrafilters with many immediate successors in this hierarchy and find positions of products of ultrafilters.
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  26. Pooling Modalities and Pointwise Intersection: Axiomatization and Decidability.Frederik Van De Putte & Dominik Klein - 2021 - Studia Logica 109 (1):47-93.
    We establish completeness and the finite model property for logics featuring the pooling modalities that were introduced in Van De Putte and Klein. The definition of our canonical models combines standard techniques with a so-called “puzzle piece construction”, which we first illustrate informally. After that, we apply it to the weakest classical logics with pooling modalities and investigate the technique’s potential for the axiomatization of stronger logics, obtained by imposing well-known frame conditions on the models.
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  27. Metrically Homogeneous Graphs of Diameter 3.Daniela A. Amato, Gregory Cherlin & H. Dugald Macpherson - 2020 - Journal of Mathematical Logic 21 (1):2050020.
    We classify countable metrically homogeneous graphs of diameter 3.
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  28. On Isomorphism Classes of Computably Enumerable Equivalence Relations.Uri Andrews & Serikzhan A. Badaev - 2020 - Journal of Symbolic Logic 85 (1):61-86.
    We examine how degrees of computably enumerable equivalence relations under computable reduction break down into isomorphism classes. Two ceers are isomorphic if there is a computable permutation of ω which reduces one to the other. As a method of focusing on nontrivial differences in isomorphism classes, we give special attention to weakly precomplete ceers. For any degree, we consider the number of isomorphism types contained in the degree and the number of isomorphism types of weakly precomplete ceers contained in the (...)
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  29. The Theory of Ceers Computes True Arithmetic.Uri Andrews, Noah Schweber & Andrea Sorbi - 2020 - Annals of Pure and Applied Logic 171 (8):102811.
    We show that the theory of the partial order of computably enumerable equivalence relations (ceers) under computable reduction is 1-equivalent to true arithmetic. We show the same result for the structure comprised of the dark ceers and the structure comprised of the light ceers. We also show the same for the structure of L-degrees in the dark, light, or complete structure. In each case, we show that there is an interpretable copy of (N, +, \times) .
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  30. The Determined Property of Baire in Reverse Math.Eric P. Astor, Damir Dzhafarov, Antonio Montalbán, Reed Solomon & Linda Brown Westrick - 2020 - Journal of Symbolic Logic 85 (1):166-198.
    We define the notion of a completely determined Borel code in reverse mathematics, and consider the principle $CD - PB$, which states that every completely determined Borel set has the property of Baire. We show that this principle is strictly weaker than $AT{R_0}$. Any ω-model of $CD - PB$ must be closed under hyperarithmetic reduction, but $CD - PB$ is not a theory of hyperarithmetic analysis. We show that whenever $M \subseteq {2^\omega }$ is the second-order part of an ω-model (...)
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  31. Many-Valued Logics. A Mathematical and Computational Introduction.Luis M. Augusto - 2020 - London: College Publications.
    2nd edition. Many-valued logics are those logics that have more than the two classical truth values, to wit, true and false; in fact, they can have from three to infinitely many truth values. This property, together with truth-functionality, provides a powerful formalism to reason in settings where classical logic—as well as other non-classical logics—is of no avail. Indeed, originally motivated by philosophical concerns, these logics soon proved relevant for a plethora of applications ranging from switching theory to cognitive modeling, and (...)
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  32. Computational Logic. Vol. 1: Classical Deductive Computing with Classical Logic. 2nd Ed.Luis M. Augusto - 2020 - London: College Publications.
    This is the 2nd edition of Computational logic. Vol. 1: Classical deductive computing with classical logic. This edition has a wholly new chapter on Datalog, a hard nut to crack from the viewpoint of semantics when negation is included.
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  33. Logical Consequences. Theory and Applications: An Introduction.Luis M. Augusto - 2020 - London: College Publications.
    2nd edition. The theory of logical consequence is central in modern logic and its applications. However, it is mostly dispersed in an abundance of often difficultly accessible papers, and rarely treated with applications in mind. This book collects the most fundamental aspects of this theory and offers the reader the basics of its applications in computer science, artificial intelligence, and cognitive science, to name but the most important fields where this notion finds its many applications.
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  34. A Lindström Theorem for Intuitionistic Propositional Logic.Guillermo Badia - 2020 - Notre Dame Journal of Formal Logic 61 (1):11-30.
    We show that propositional intuitionistic logic is the maximal abstract logic satisfying a certain form of compactness, the Tarski union property, and preservation under asimulations.
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  35. Structure and Representation of Semimodules Over Inclines.Ruiqi Bai & Yichuan Yang - 2020 - Annals of Pure and Applied Logic 171 (10):102844.
    An incline S is a commutative semiring where r+1=1 for any r \in S . We note that the ideal lattice of an S-semimodule is naturally an S-semimodule and so is its congruence lattice when S is transitive. We prove that the categories of complete S-semimodules, together with dual functor, internal hom and tensor product, is a ⋆-autonomous category. We define the locally and globally maximal congruences which are related to Birkhoff subdirect product decomposition. We show that the categories of (...)
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  36. Forcing and the Universe of Sets: Must We Lose Insight?Neil Barton - 2020 - Journal of Philosophical Logic 49 (4):575-612.
    A central area of current philosophical debate in the foundations of mathematics concerns whether or not there is a single, maximal, universe of set theory. Universists maintain that there is such a universe, while Multiversists argue that there are many universes, no one of which is ontologically privileged. Often forcing constructions that add subsets to models are cited as evidence in favour of the latter. This paper informs this debate by analysing ways the Universist might interpret this discourse that seems (...)
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  37. Inner-Model Reflection Principles.Neil Barton, Andrés Eduardo Caicedo, Gunter Fuchs, Joel David Hamkins, Jonas Reitz & Ralf Schindler - 2020 - Studia Logica 108 (3):573-595.
    We introduce and consider the inner-model reflection principle, which asserts that whenever a statement \varphi(a) in the first-order language of set theory is true in the set-theoretic universe V, then it is also true in a proper inner model W \subset A. A stronger principle, the ground-model reflection principle, asserts that any such \varphi(a) true in V is also true in some non-trivial ground model of the universe with respect to set forcing. These principles each express a form of width (...)
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  38. Assigning an Isomorphism Type to a Hyperdegree.Howard Becker - 2020 - Journal of Symbolic Logic 85 (1):325-337.
    Let L be a computable vocabulary, let X_L be the space of L-structures with universe ω and let f:2^\omega \rightarrow X_L be a hyperarithmetic function such that for all x,y \in 2^\omega, if x \equiv _h y then f(x) \cong f(y). One of the following two properties must hold. (1) The Scott rank of f(0) is \omega _1^{CK} + 1. (2) For all x \in 2^\omega, f(x) \cong f(0).
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  39. Positive Amalgamation.Mohammed Belkasmi - 2020 - Logica Universalis 14 (2):243-258.
    We study the amalgamation property in positive logic, where we shed light on some connections between the amalgamation property, Robinson theories, model-complete theories and the Hausdorff property.
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  40. Algebraically Closed Structures in Positive Logic.Mohammed Belkasmi - 2020 - Annals of Pure and Applied Logic 171 (9):102822.
    In this paper we extend of the notion of algebraically closed given in the case of groups and skew fields to an arbitrary h-inductive theory. The main subject of this paper is the study of the notion of positive algebraic closedness and its relationship with the notion of positive closedness and the amalgamation property.
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  41. Isometry Groups of Borel Randomizations.Alexander Berenstein & Rafael Zamora - 2020 - Notre Dame Journal of Formal Logic 61 (2):297-316.
    We study global dynamical properties of the isometry group of the Borel randomization of a separable complete structure. We show that if properties such as the Rokhlin property, topometric generics, and extreme amenability hold for the isometry group of the structure, then they also hold in the isometry group of the randomization.
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  42. Weaker variants of infinite time Turing machines.Matteo Bianchetti - 2020 - Archive for Mathematical Logic 59 (3-4):335-365.
    Infinite time Turing machines represent a model of computability that extends the operations of Turing machines to transfinite ordinal time by defining the content of each cell at limit steps to be the lim sup of the sequences of previous contents of that cell. In this paper, we study a computational model obtained by replacing the lim sup rule with an ‘eventually constant’ rule: at each limit step, the value of each cell is defined if and only if the content (...)
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  43. Functional Representation of Finitely Generated Free Algebras in Subvarieties of BL-Algebras.Manuela Busaniche, José Luis Castiglioni & Noemí Lubomirsky - 2020 - Annals of Pure and Applied Logic 171 (2):102757.
    Consider any subvariety of BL-algebras generated by a single BL-chain which is the ordinal sum of the standard MV-algebra on [0, 1] and a basic hoop H. We present a geometrical characterization of elements in the finitely generated free algebra of each of these subvarieties. In this characterization there is a clear insight of the role of the regular and dense elements of the generating chain. As an application, we analyze maximal and prime filters in the free algebra.
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  44. Expander Construction in VNC1.Sam Buss, Valentine Kabanets, Antonina Kolokolova & Michal Koucký - 2020 - Annals of Pure and Applied Logic 171 (7):102796.
    We give a combinatorial analysis (using edge expansion) of a variant of the iterative expander construction due to Reingold, Vadhan, and Wigderson [44], and show that this analysis can be formalized in the bounded arithmetic system VNC^1 (corresponding to the “NC^1 reasoning”). As a corollary, we prove the assumption made by Jeřábek [28] that a construction of certain bipartite expander graphs can be formalized in VNC^1 . This in turn implies that every proof in Gentzen's sequent calculus LK of a (...)
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  45. Polish Metric Spaces with Fixed Distance Set.Riccardo Camerlo, Alberto Marcone & Luca Motto Ros - 2020 - Annals of Pure and Applied Logic 171 (10):102832.
    We study Polish spaces for which a set of possible distances $A \subseteq R^+$ is fixed in advance. We determine, depending on the properties of A, the complexity of the collection of all Polish metric spaces with distances in A, obtaining also example of sets in some Wadge classes where not many natural examples are known. Moreover we describe the properties that A must have in order that all Polish spaces with distances in that set belong to a given class, (...)
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  46. Every Zero-Dimensional Homogeneous Space is Strongly Homogeneous Under Determinacy.Raphaël Carroy, Andrea Medini & Sandra Müller - 2020 - Journal of Mathematical Logic 20 (3):2050015.
    All spaces are assumed to be separable and metrizable. We show that, assuming the Axiom of Determinacy, every zero-dimensional homogeneous space is strongly homogeneous (i.e. all its non-empty clopen subspaces are homeomorphic), with the trivial exception of locally compact spaces. In fact, we obtain a more general result on the uniqueness of zero-dimensional homogeneous spaces which generate a given Wadge class. This extends work of van Engelen (who obtained the corresponding results for Borel spaces), complements a result of van Douwen, (...)
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  47. Remarks on Generic Stability in Independent Theories.Gabriel Conant & Kyle Gannon - 2020 - Annals of Pure and Applied Logic 171 (2):102736.
    In NIP theories, generically stable Keisler measures can be characterized in several ways. We analyze these various forms of “generic stability” in arbitrary theories. Among other things, we show that the standard definition of generic stability for types coincides with the notion of a frequency interpretation measure. We also give combinatorial examples of types in NSOP theories that are finitely approximated but not generically stable, as well as ϕ-types in simple theories that are definable and finitely satisfiable in a small (...)
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  48. Defining Integer-Valued Functions in Rings of Continuous Definable Functions Over a Topological Field.Luck Darnière & Marcus Tressl - 2020 - Journal of Mathematical Logic 20 (3):2050014.
    Let [Formula: see text] be an expansion of either an ordered field [Formula: see text], or a valued field [Formula: see text]. Given a definable set [Formula: see text] let [Formula: see text] be the ring of continuous definable functions from [Formula: see text] to [Formula: see text]. Under very mild assumptions on the geometry of [Formula: see text] and on the structure [Formula: see text], in particular when [Formula: see text] is [Formula: see text]-minimal or [Formula: see text]-minimal, or (...)
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  49. Some Lower Bounds on Shelah Rank in the Free Group.Javier de la Nuez González, Chloé Perin & Rizos Sklinos - 2020 - Annals of Pure and Applied Logic 171 (6):102794.
    We give some lower bounds on the Shelah rank of varieties in the free group whose coordinate groups are hyperbolic towers.
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  50. The Ramsey Theory of the Universal Homogeneous Triangle-Free Graph.Natasha Dobrinen - 2020 - Journal of Mathematical Logic 20 (2):2050012.
    The universal homogeneous triangle-free graph, constructed by Henson [A family of countable homogeneous graphs, Pacific J. Math.38(1) (1971) 69–83] and denoted H3, is the triangle-free analogue of the Rado graph. While the Ramsey theory of the Rado graph has been completely established, beginning with Erdős–Hajnal–Posá [Strong embeddings of graphs into coloured graphs, in Infinite and Finite Sets. Vol.I, eds. A. Hajnal, R. Rado and V. Sós, Colloquia Mathematica Societatis János Bolyai, Vol. 10 (North-Holland, 1973), pp. 585–595] and culminating in work (...)
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