Set Theory

Edited by Toby Meadows (University of California, Irvine)
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  1. Schopenhauers Logikdiagramme in den Mathematiklehrbüchern Adolph Diesterwegs.Jens Lemanski - 2022 - Siegener Beiträge Zur Geschichte Und Philosophie der Mathematik 16:97-127.
    Ein Beispiel für die Rezeption und Fortführung der schopenhauerschen Logik findet man in den Mathematiklehrbüchern Friedrich Adolph Wilhelm Diesterwegs (1790–1866), In diesem Aufsatz werden die historische und systematische Dimension dieser Anwendung von Logikdiagramme auf die Mathematik skizziert. In Kapitel 2 wird zunächst die frühe Rezeption der schopenhauerschen Logik und Philosophie der Mathematik vorgestellt. Dabei werden einige oftmals tradierte Vorurteile, die das Werk Schopenhauers betreffen, in Frage gestellt oder sogar ausgeräumt. In Kapitel 3 wird dann die Philosophie der Mathematik und der (...)
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  2. Reduction games, provability and compactness.Damir D. Dzhafarov, Denis R. Hirschfeldt & Sarah Reitzes - 2022 - Journal of Mathematical Logic 22 (3).
    Journal of Mathematical Logic, Volume 22, Issue 03, December 2022. Hirschfeldt and Jockusch (2016) introduced a two-player game in which winning strategies for one or the other player precisely correspond to implications and non-implications between [math] principles over [math]-models of [math]. They also introduced a version of this game that similarly captures provability over [math]. We generalize and extend this game-theoretic framework to other formal systems, and establish a certain compactness result that shows that if an implication [math] between two (...)
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  3. Sigma-Prikry forcing II: Iteration Scheme.Alejandro Poveda, Assaf Rinot & Dima Sinapova - 2022 - Journal of Mathematical Logic 22 (3):2150019.
    In Part I of this series [A. Poveda, A. Rinot and D. Sinapova, Sigma-Prikry forcing I: The axioms, Canad. J. Math. 73(5) (2021) 1205–1238], we introduced a class of notions of forcing which we call [Formula: see text]-Prikry, and showed that many of the known Prikry-type notions of forcing that center around singular cardinals of countable cofinality are [Formula: see text]-Prikry. We showed that given a [Formula: see text]-Prikry poset [Formula: see text] and a [Formula: see text]-name for a non-reflecting (...)
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  4. Mathematical Modality: An Investigation of Set Theoretic Contingency.Andrew Bacon -
    An increasing amount of contemporary philosophy of mathematics posits, and theorizes in terms of special kinds of mathematical modality. The goal of this paper is to bring recent work on higher-order metaphysics to bear on the investigation of these modalities. -/- The main focus of the paper will be views that posit mathematical contingency or indeterminacy about statements that concern the `width' of the set theoretic universe, such as Cantor's continuum hypothesis. In the higher-order framework I show that contingency about (...)
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  5. Lower and Upper Estimates of the Quantity of Algebraic Numbers.Yaroslav Sergeyev - 2023 - Mediterranian Journal of Mathematics 20:12.
    It is well known that the set of algebraic numbers (let us call it A) is countable. In this paper, instead of the usage of the classical terminology of cardinals proposed by Cantor, a recently introduced methodology using ①-based infinite numbers is applied to measure the set A (where the number ① is called grossone). Our interest to this methodology is explained by the fact that in certain cases where cardinals allow one to say only whether a set is countable (...)
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  6. Extension of Soft Set to Hypersoft Set, and then to Plithogenic Hypersoft Set.Florentin Smarandache - 2018 - Neutrosophic Sets and Systems 22 (1):168-170.
    In this paper, we generalize the soft set to the hypersoft set by transforming the function F into a multi-attribute function. Then we introduce the hybrids of Crisp, Fuzzy, Intuitionistic Fuzzy, Neutrosophic, and Plithogenic Hypersoft Set.
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  7. Introduction to the Complex Refined Neutrosophic Set.Florentin Smarandache - 2017 - Critical Review: A Journal of Politics and Society 14 (1):5-9.
    In this paper, one extends the single-valued complex neutrosophic set to the subsetvalued complex neutrosophic set, and afterwards to the subset-valued complex refined neutrosophic set.
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  8. Realizing realizability results with classical constructions.Asaf Karagila - 2019 - Bulletin of Symbolic Logic 25 (4):429-445.
    J. L. Krivine developed a new method based on realizability to construct models of set theory where the axiom of choice fails. We attempt to recreate his results in classical settings, i.e., symmetric extensions. We also provide a new condition for preserving well ordered, and other particular type of choice, in the general settings of symmetric extensions.
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  9. MORRIS, Sean: Quine, New Foundations, and the Philosophy of Set Theory.Ádám Tamas Tuboly - 2019 - Filozofia 74 (6).
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  10. RETRACTED ARTICLE: The Twin Primes Conjecture is True in the Standard Model of Peano Arithmetic: Applications of Rasiowa–Sikorski Lemma in Arithmetic (I).Janusz Czelakowski - forthcoming - Studia Logica:1-1.
    The paper is concerned with the old conjecture that there are infinitely many twin primes. In the paper we show that this conjecture is true, that is, it is true in the standard model of arithmetic. The proof is based on Rasiowa–Sikorski Lemma. The key role are played by the derived notion of a Rasiowa–Sikorski set and the method of forcing adjusted to arbitrary first–order languages. This approach was developed in the papers Czelakowski [ 4, 5 ]. The central idea (...)
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  11. An axiomatic approach to forcing in a general setting.Rodrigo A. Freire & Peter Holy - 2022 - Bulletin of Symbolic Logic 28 (3):427-450.
    The technique of forcing is almost ubiquitous in set theory, and it seems to be based on technicalities like the concepts of genericity, forcing names and their evaluations, and on the recursively defined forcing predicates, the definition of which is particularly intricate for the basic case of atomic first order formulas. In his [3], the first author has provided an axiomatic framework for set forcing over models of $\mathrm {ZFC}$ that is a collection of guiding principles for extensions over which (...)
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  12. Four cardinals and their relations in ZF.Lorenz Halbeisen, Riccardo Plati, Salome Schumacher & Saharon Shelah - 2023 - Annals of Pure and Applied Logic 174 (2):103200.
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  13. Model-Theoretic Properties of Dynamics on the Cantor Set.Christopher J. Eagle & Alan Getz - 2022 - Notre Dame Journal of Formal Logic 63 (3):357-371.
    We examine topological dynamical systems on the Cantor set from the point of view of the continuous model theory of commutative C*-algebras. After some general remarks, we focus our attention on the generic homeomorphism of the Cantor set, as constructed by Akin, Glasner, and Weiss. We show that this homeomorphism is the prime model of its theory. We also show that the notion of “generic” used by Akin, Glasner, and Weiss is distinct from the notion of “generic” encountered in Fraïssé (...)
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  14. Ontology and Arbitrariness.David Builes - 2022 - Australasian Journal of Philosophy 100 (3):485-495.
    In many different ontological debates, anti-arbitrariness considerations push one towards two opposing extremes. For example, in debates about mereology, one may be pushed towards a maximal ontology (mereological universalism) or a minimal ontology (mereological nihilism), because any intermediate view seems objectionably arbitrary. However, it is usually thought that anti-arbitrariness considerations on their own cannot decide between these maximal or minimal views. I will argue that this is a mistake. Anti-arbitrariness arguments may be used to motivate a certain popular thesis in (...)
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  15. A Reassessment of Cantorian Abstraction based on the $$\varepsilon $$ ε -operator.Nicola Bonatti - 2022 - Synthese 200 (5):1-26.
    Cantor’s abstractionist account of cardinal numbers has been criticized by Frege as a psychological theory of numbers which leads to contradiction. The aim of the paper is to meet these objections by proposing a reassessment of Cantor’s proposal based upon the set theoretic framework of Bourbaki—called BK—which is a First-order set theory extended with Hilbert’s \-operator. Moreover, it is argued that the BK system and the \-operator provide a faithful reconstruction of Cantor’s insights on cardinal numbers. I will introduce first (...)
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  16. Subcompact Cardinals, Type Omission, and Ladder Systems.Yair Hayut & Menachem Magidor - 2022 - Journal of Symbolic Logic 87 (3):1111-1129.
    We provide a model theoretical and tree property-like characterization of $\lambda $ - $\Pi ^1_1$ -subcompactness and supercompactness. We explore the behavior of these combinatorial principles at accessible cardinals.
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  17. Complexity of Index Sets of Descriptive Set-Theoretic Notions.Reese Johnston & Dilip Raghavan - 2022 - Journal of Symbolic Logic 87 (3):894-911.
    Descriptive set theory and computability theory are closely-related fields of logic; both are oriented around a notion of descriptive complexity. However, the two fields typically consider objects of very different sizes; computability theory is principally concerned with subsets of the naturals, while descriptive set theory is interested primarily in subsets of the reals. In this paper, we apply a generalization of computability theory, admissible recursion theory, to consider the relative complexity of notions that are of interest in descriptive set theory. (...)
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  18. Meager-Additive Sets in Topological Groups.Ondřej Zindulka - 2022 - Journal of Symbolic Logic 87 (3):1046-1064.
    By the Galvin–Mycielski–Solovay theorem, a subset X of the line has Borel’s strong measure zero if and only if $M+X\neq \mathbb {R}$ for each meager set M.A set $X\subseteq \mathbb {R}$ is meager-additive if $M+X$ is meager for each meager set M. Recently a theorem on meager-additive sets that perfectly parallels the Galvin–Mycielski–Solovay theorem was proven: A set $X\subseteq \mathbb {R}$ is meager-additive if and only if it has sharp measure zero, a notion akin to strong measure zero.We investigate the (...)
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  19. Descriptive Complexity in Cantor Series.Dylan Airey, Steve Jackson & Bill Mance - 2022 - Journal of Symbolic Logic 87 (3):1023-1045.
    A Cantor series expansion for a real number x with respect to a basic sequence $Q=(q_1,q_2,\dots )$, where $q_i \geq 2$, is a generalization of the base b expansion to an infinite sequence of bases. Ki and Linton in 1994 showed that for ordinary base b expansions the set of normal numbers is a $\boldsymbol {\Pi }^0_3$ -complete set, establishing the exact complexity of this set. In the case of Cantor series there are three natural notions of normality: normality, ratio (...)
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  20. The Turing Degrees and Keisler’s Order.Maryanthe Malliaris & Saharon Shelah - forthcoming - Journal of Symbolic Logic:1-11.
    There is a Turing functional $\Phi $ taking $A^\prime $ to a theory $T_A$ whose complexity is exactly that of the jump of A, and which has the property that $A \leq _T B$ if and only if $T_A \trianglelefteq T_B$ in Keisler’s order. In fact, by more elaborate means and related theories, we may keep the complexity at the level of A without using the jump.
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  21. Strong Compactness, Square, Gch, and Woodin Cardinals.Arthur W. Apter - forthcoming - Journal of Symbolic Logic:1-9.
    We show the consistency, relative to the appropriate supercompactness or strong compactness assumptions, of the existence of a non-supercompact strongly compact cardinal $\kappa _0$ (the least measurable cardinal) exhibiting properties which are impossible when $\kappa _0$ is supercompact. In particular, we construct models in which $\square _{\kappa ^+}$ holds for every inaccessible cardinal $\kappa $ except $\kappa _0$, GCH fails at every inaccessible cardinal except $\kappa _0$, and $\kappa _0$ is less than the least Woodin cardinal.
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  22. Mathematical Progress — On Maddy and Beyond.Simon Weisgerber - forthcoming - Philosophia Mathematica.
    A key question of the ‘maverick’ tradition of the philosophy of mathematical practice is addressed, namely what is mathematical progress. The investigation is based on an article by Penelope Maddy devoted to this topic in which she considers only contributions ‘of some mathematical importance’ as progress. With the help of a case study from contemporary mathematics, more precisely from tropical geometry, a few issues with her proposal are identified. Taking these issues into consideration, an alternative account of ‘mathematical importance’, broadly (...)
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  23. Cohen-like first order structures.Ziemowit Kostana - 2023 - Annals of Pure and Applied Logic 174 (1):103172.
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  24. Non-stationary support iterations of Prikry forcings and restrictions of ultrapower embeddings to the ground model.Moti Gitik & Eyal Kaplan - 2023 - Annals of Pure and Applied Logic 174 (1):103164.
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  25. The poset of all logics II: Leibniz classes and hierarchy.R. Jansana & T. Moraschini - forthcoming - Journal of Symbolic Logic:1-39.
    A Leibniz class is a class of logics closed under the formation of term-equivalent logics, compatible expansions, and non-indexed products of sets of logics. We study the complete lattice of all Leibniz classes, called the Leibniz hierarchy. In particular, it is proved that the classes of truth-equational and assertional logics are meet-prime in the Leibniz hierarchy, while the classes of protoalgebraic and equivalential logics are meet-reducible. However, the last two classes are shown to be determined by Leibniz conditions consisting of (...)
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  26. Incompatible bounded category forcing axioms.David Asperó & Matteo Viale - 2022 - Journal of Mathematical Logic 22 (2).
    Journal of Mathematical Logic, Volume 22, Issue 02, August 2022. We introduce bounded category forcing axioms for well-behaved classes [math]. These are strong forms of bounded forcing axioms which completely decide the theory of some initial segment of the universe [math] modulo forcing in [math], for some cardinal [math] naturally associated to [math]. These axioms naturally extend projective absoluteness for arbitrary set-forcing — in this situation [math] — to classes [math] with [math]. Unlike projective absoluteness, these higher bounded category forcing (...)
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  27. Forcing the [math]-separation property.Stefan Hoffelner - 2022 - Journal of Mathematical Logic 22 (2).
    Journal of Mathematical Logic, Volume 22, Issue 02, August 2022. We generically construct a model in which the [math]-separation property is true, i.e. every pair of disjoint [math]-sets can be separated by a [math]-definable set. This answers an old question from the problem list “Surrealist landscape with figures” by A. Mathias from 1968. We also construct a model in which the (lightface) [math]-separation property is true.
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  28. Structural reflection, shrewd cardinals and the size of the continuum.Philipp Lücke - 2022 - Journal of Mathematical Logic 22 (2).
    Journal of Mathematical Logic, Volume 22, Issue 02, August 2022. Motivated by results of Bagaria, Magidor and Väänänen, we study characterizations of large cardinal properties through reflection principles for classes of structures. More specifically, we aim to characterize notions from the lower end of the large cardinal hierarchy through the principle [math] introduced by Bagaria and Väänänen. Our results isolate a narrow interval in the large cardinal hierarchy that is bounded from below by total indescribability and from above by subtleness, (...)
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  29. The permutations with N_ non-fixed points and the sequences with length _N of a set.Jukkrid Nuntasri & Pimpen Vejjajiva - forthcoming - Journal of Symbolic Logic:1-10.
    We write $\mathcal {S}_n(A)$ for the set of permutations of a set A with n non-fixed points and $\mathrm {{seq}}^{1-1}_n(A)$ for the set of one-to-one sequences of elements of A with length n where n is a natural number greater than $1$. With the Axiom of Choice, $|\mathcal {S}_n(A)|$ and $|\mathrm {{seq}}^{1-1}_n(A)|$ are equal for all infinite sets A. Among our results, we show, in ZF, that $|\mathcal {S}_n(A)|\leq |\mathrm {{seq}}^{1-1}_n(A)|$ for any infinite set A if ${\mathrm {AC}}_{\leq n}$ is (...)
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  30. Constructive strong regularity and the extension property of a compactification.Giovanni Curi - 2023 - Annals of Pure and Applied Logic 174 (1):103154.
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  31. Extensional Realizability and Choice for Dependent Types in Intuitionistic Set Theory.Emanuele Frittaion - forthcoming - Journal of Symbolic Logic:1-32.
    In [17], we introduced an extensional variant of generic realizability [22], where realizers act extensionally on realizers, and showed that this form of realizability provides inner models of $\mathsf {CZF}$ (constructive Zermelo–Fraenkel set theory) and $\mathsf {IZF}$ (intuitionistic Zermelo–Fraenkel set theory), that further validate $\mathsf {AC}_{\mathsf {FT}}$ (the axiom of choice in all finite types). In this paper, we show that extensional generic realizability validates several choice principles for dependent types, all exceeding $\mathsf {AC}_{\mathsf {FT}}$. We then show that adding (...)
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  32. Correction to: The Hahn Embedding Theorem for a Class of Residuated Semigroups.Sándor Jenei - 2022 - Studia Logica 110 (4):1135-1135.
    A Correction to this paper has been published: 10.1007/s11225-020-09933-y.
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  33. Group Representation for Even and Odd Involutive Commutative Residuated Chains.Sándor Jenei - 2022 - Studia Logica 110 (4):881-922.
    For odd and for even involutive, commutative residuated chains a representation theorem is presented in this paper by means of direct systems of abelian o-groups equipped with further structure. This generalizes the corresponding result of J. M. Dunnabout finite Sugihara monoids.
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  34. The spectrum of independence, II.Vera Fischer & Saharon Shelah - 2022 - Annals of Pure and Applied Logic 173 (9):103161.
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  35. A Universal Algebraic Set Theory Built on Mereology with Applications.Ioachim Drugus - 2022 - Logica Universalis 16 (1):253-283.
    Category theory is often treated as an algebraic foundation for mathematics, and the widely known algebraization of ZF set theory in terms of this discipline is referenced as “categorical set theory” or “set theory for category theory”. The method of algebraization used in this theory has not been formulated in terms of universal algebra so far. In current paper, a _universal algebraic_ method, i.e. one formulated in terms of universal algebra, is presented and used for algebraization of a ground mereological (...)
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  36. Characterizing existence of certain ultrafilters.Rafał Filipów, Krzysztof Kowitz & Adam Kwela - 2022 - Annals of Pure and Applied Logic 173 (9):103157.
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  37. Set theoretical analogues of the Barwise-Schlipf theorem.Ali Enayat - 2022 - Annals of Pure and Applied Logic 173 (9):103158.
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  38. P-points, MAD families and Cardinal Invariants.Osvaldo Guzmán González - 2022 - Bulletin of Symbolic Logic 28 (2):258-260.
    The main topics of this thesis are cardinal invariants, P -points and MAD families. Cardinal invariants of the continuum are cardinal numbers that are bigger than $\aleph _{0}$ and smaller or equal than $\mathfrak {c}.$ Of course, they are only interesting when they have some combinatorial or topological definition. An almost disjoint family is a family of infinite subsets of $\omega $ such that the intersection of any two of its elements is finite. A MAD family is a maximal almost (...)
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  39. Order types of models of fragments of peano arithmetic.Lorenzo Galeotti & Benedikt Löwe - 2022 - Bulletin of Symbolic Logic 28 (2):182-206.
    The complete characterisation of order types of non-standard models of Peano arithmetic and its extensions is a famous open problem. In this paper, we consider subtheories of Peano arithmetic, in particular, theories formulated in proper fragments of the full language of arithmetic. We study the order types of their non-standard models and separate all considered theories via their possible order types. We compare the theories with and without induction and observe that the theories without induction tend to have an algebraic (...)
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  40. Exact saturation in pseudo-elementary classes for simple and stable theories.Itay Kaplan, Nicholas Ramsey & Saharon Shelah - forthcoming - Journal of Mathematical Logic.
    We use exact saturation to study the complexity of unstable theories, showing that a variant of this notion called pseudo-elementary class (PC)-exact saturation meaningfully reflects combinatorial dividing lines. We study PC-exact saturation for stable and simple theories. Among other results, we show that PC-exact saturation characterizes the stability cardinals of size at least continuum of a countable stable theory and, additionally, that simple unstable theories have PC-exact saturation at singular cardinals satisfying mild set-theoretic hypotheses. This had previously been open even (...)
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  41. Compactness and guessing principles in the Radin extensions.Omer Ben-Neria & Jing Zhang - forthcoming - Journal of Mathematical Logic.
    We investigate the interaction between compactness principles and guessing principles in the Radin forcing extensions. In particular, we show that in any Radin forcing extension with respect to a measure sequence on [Formula: see text], if [Formula: see text] is weakly compact, then [Formula: see text] holds. This provides contrast with a well-known theorem of Woodin, who showed that in a certain Radin extension over a suitably prepared ground model relative to the existence of large cardinals, the diamond principle fails (...)
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  42. Compactness versus hugeness at successor cardinals.Sean Cox & Monroe Eskew - forthcoming - Journal of Mathematical Logic.
    If [Formula: see text] is regular and [Formula: see text], then the existence of a weakly presaturated ideal on [Formula: see text] implies [Formula: see text]. This partially answers a question of Foreman and Magidor about the approachability ideal on [Formula: see text]. As a corollary, we show that if there is a presaturated ideal [Formula: see text] on [Formula: see text] such that [Formula: see text] is semiproper, then CH holds. We also show some barriers to getting the tree (...)
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  43. Retraction – measuring club-sequences together with the continuum large.David Asperó & Miguel Angel Mota - 2022 - Journal of Symbolic Logic 87 (2):870-870.
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  44. The σ1-definable universal finite sequence.Joel David Hamkins & Kameryn J. Williams - 2022 - Journal of Symbolic Logic 87 (2):783-801.
    We introduce the $\Sigma _1$ -definable universal finite sequence and prove that it exhibits the universal extension property amongst the countable models of set theory under end-extension. That is, the sequence is $\Sigma _1$ -definable and provably finite; the sequence is empty in transitive models; and if M is a countable model of set theory in which the sequence is s and t is any finite extension of s in this model, then there is an end-extension of M to a (...)
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  45. Variations on determinacy and ℵω1.Ramez L. Sami - 2022 - Journal of Symbolic Logic 87 (2):721-731.
    We consider a seemingly weaker form of $\Delta ^{1}_{1}$ Turing determinacy.Let $2 \leqslant \rho < \omega _{1}^{\mathsf {CK}}$, $\textrm{Weak-Turing-Det}_{\rho }$ is the statement:Every $\Delta ^{1}_{1}$ set of reals cofinal in the Turing degrees contains two Turing distinct, $\Delta ^{0}_{\rho }$ -equivalent reals.We show in $\mathsf {ZF}^-$ : $\textrm{Weak-Turing-Det}_{\rho }$ implies: for every $\nu < \omega _{1}^{\mathsf {CK}}$ there is a transitive model ${M \models \mathsf {ZF}^{-} + \textrm{``}\aleph _{\nu } \textrm{ exists''.}}$ As a corollary:If every cofinal $\Delta ^{1}_{1}$ set of (...)
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  46. Infinite strings and their large scale properties.Bakh Khoussainov & Toru Takisaka - 2022 - Journal of Symbolic Logic 87 (2):585-625.
    The aim of this paper is to shed light on our understanding of large scale properties of infinite strings. We say that one string $\alpha $ has weaker large scale geometry than that of $\beta $ if there is color preserving bi-Lipschitz map from $\alpha $ into $\beta $ with small distortion. This definition allows us to define a partially ordered set of large scale geometries on the classes of all infinite strings. This partial order compares large scale geometries of (...)
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  47. Identity crisis between supercompactness and vǒpenka’s principle.Yair Hayut, Menachem Magidor & Alejandro Poveda - 2022 - Journal of Symbolic Logic 87 (2):626-648.
    In this paper we study the notion of $C^{}$ -supercompactness introduced by Bagaria in [3] and prove the identity crises phenomenon for such class. Specifically, we show that consistently the least supercompact is strictly below the least $C^{}$ -supercompact but also that the least supercompact is $C^{}$ -supercompact }$ -supercompact). Furthermore, we prove that under suitable hypothesis the ultimate identity crises is also possible. These results solve several questions posed by Bagaria and Tsaprounis.
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  48. Asymptotic analysis of skolem’s exponential functions.Alessandro Berarducci & Marcello Mamino - 2022 - Journal of Symbolic Logic 87 (2):758-782.
    Skolem studied the germs at infinity of the smallest class of real valued functions on the positive real line containing the constant $1$, the identity function ${\mathbf {x}}$, and such that whenever f and g are in the set, $f+g,fg$ and $f^g$ are in the set. This set of germs is well ordered and Skolem conjectured that its order type is epsilon-zero. Van den Dries and Levitz computed the order type of the fragment below $2^{2^{\mathbf {x}}}$. Here we prove that (...)
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  49. On categoricity in successive cardinals.Sebastien Vasey - 2022 - Journal of Symbolic Logic 87 (2):545-563.
    We investigate, in ZFC, the behavior of abstract elementary classes categorical in many successive small cardinals. We prove for example that a universal $\mathbb {L}_{\omega _1, \omega }$ sentence categorical on an end segment of cardinals below $\beth _\omega $ must be categorical also everywhere above $\beth _\omega $. This is done without any additional model-theoretic hypotheses and generalizes to the much broader framework of tame AECs with weak amalgamation and coherent sequences.
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  50. Typicality à la Russell in Set Theory.Athanassios Tzouvaras - 2022 - Notre Dame Journal of Formal Logic 63 (2).
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