Set Theory

Edited by Toby Meadows (University of California, Irvine)
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  1. What the Tortoise Said to Achilles: Lewis Carroll’s Paradox in Terms of Hilbert Arithmetic.Vasil Penchev - 2021 - Logic and Philosophy of Mathematics eJournal (Elsevier: SSRN) 13 (22):1-32.
    Lewis Carroll, both logician and writer, suggested a logical paradox containing furthermore two connotations (connotations or metaphors are inherent in literature rather than in mathematics or logics). The paradox itself refers to implication demonstrating that an intermediate implication can be always inserted in an implication therefore postponing its ultimate conclusion for the next step and those insertions can be iteratively and indefinitely added ad lib, as if ad infinitum. Both connotations clear up links due to the shared formal structure with (...)
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  2. Reflection Principles and Second-Order Choice Principles with Urelements.Bokai Yao - 2022 - Annals of Pure and Applied Logic 173 (4):103073.
    We study reflection principles in Kelley-Morse set theory with urelements (KMU). We first show that First-Order Reflection Principle is not provable in KMU with Global Choice. We then show that KMU + Limitation of Size + Second-Order Reflection Principle is mutually interpretable with KM + Second-Order Reflection Principle. Furthermore, these two theories are also shown to be bi-interpretable with parameters. Finally, assuming the existence of a κ+-supercompact cardinal κ in KMU, we construct a model of KMU + Second-Order Reflection Principle (...)
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  3. Reinterpreting the Universe-Multiverse Debate in Light of Inter-Model Inconsistency in Set Theory.Daniel Kuby - manuscript
    In this paper I apply the concept of _inter-Model Inconsistency in Set Theory_ (MIST), introduced by Carolin Antos (this volume), to select positions in the current universe-multiverse debate in philosophy of set theory: I reinterpret H. Woodin’s _Ultimate L_, J. D. Hamkins’ multiverse, S.-D. Friedman’s hyperuniverse and the algebraic multiverse as normative strategies to deal with the situation of de facto inconsistency toleration in set theory as described by MIST. In particular, my aim is to situate these positions on the (...)
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  4. Most-Intersection of Countable Sets.Ahmet Çevik & Selçuk Topal - forthcoming - Journal of Applied Non-Classical Logics:1-12.
    We introduce a novel set-intersection operator called ‘most-intersection’ based on the logical quantifier ‘most’, via natural density of countable sets, to be used in determining the majority chara...
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  5. Reinhardt Cardinals and Iterates of V.Farmer Schlutzenberg - 2022 - Annals of Pure and Applied Logic 173 (2):103056.
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  6. The Isomorphism Relation of Theories with S-DOP in the Generalised Baire Spaces.Miguel Moreno - 2022 - Annals of Pure and Applied Logic 173 (2):103044.
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  7. Canonical Fragments of the Strong Reflection Principle.Gunter Fuchs - 2021 - Journal of Mathematical Logic 21 (3):2150023.
    For an arbitrary forcing class Γ, the Γ-fragment of Todorčević’s strong reflection principle SRPis isolated in such a way that the forcing axiom for Γ implies the Γ-fragment of SRP, the sta...
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  8. Controlling Cardinal Characteristics Without Adding Reals.Martin Goldstern, Jakob Kellner, Diego A. Mejía & Saharon Shelah - 2020 - Journal of Mathematical Logic 21 (3).
    We investigate the behavior of cardinal characteristics of the reals under extensions that do not add new <κ-sequences. As an application, we show that consistently the followi...
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  9. An Unpublished Theorem of Solovay on OD Partitions of Reals Into Two Non-OD Parts, Revisited.Ali Enayat & Vladimir Kanovei - 2020 - Journal of Mathematical Logic 21 (3):2150014.
    A definable pair of disjoint non-OD sets of reals exists in the Sacks and ????0-large generic extensions of the constructible universe L. More specifically, if a∈2ω is eith...
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  10. The Weakness of the Pigeonhole Principle Under Hyperarithmetical Reductions.Benoit Monin & Ludovic Patey - 2020 - Journal of Mathematical Logic 21 (3):2150013.
    The infinite pigeonhole principle for 2-partitions asserts the existence, for every set A, of an infinite subset of A or of its complement. In this paper, we study the infinite pigeonhole pr...
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  11. Logic of Convex Order.Chenwei Shi & Yang Sun - 2021 - Studia Logica 109 (5):1019-1047.
    Based on a pre-order on a set, we axiomatize the Egli-Milner order on the power set and show how it is related to the Lewis order. Moreover, we consider a strict version of the Egli-Milner order and show how it can be related to the semantics of conditionals based on a priority structure.
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  12. Higher Dimensional Cardinal Characteristics for Sets of Functions.Corey Bacal Switzer - 2022 - Annals of Pure and Applied Logic 173 (1):103031.
  13. Against ‘Interpretation’: Quantum Mechanics Beyond Syntax and Semantics.Raoni Wohnrath Arroyo & Gilson Olegario da Silva - 2021 - Axiomathes:1-37.
    The question “what is an interpretation?” is often intertwined with the perhaps even harder question “what is a scientific theory?”. Given this proximity, we try to clarify the first question to acquire some ground for the latter. The quarrel between the syntactic and semantic conceptions of scientific theories occupied a large part of the scenario of the philosophy of science in the 20th century. For many authors, one of the two currents needed to be victorious. We endorse that such debate, (...)
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  14. A Philosophical Argument for the Beginning of Time.Laureano Luna & Jacobus Erasmus - 2020 - Prolegomena 19 (2):161-176.
    A common argument in support of a beginning of the universe used by advocates of the kalām cosmological argument (KCA) is the argument against the possibility of an actual infinite, or the “Infinity Argument”. However, it turns out that the Infinity Argument loses some of its force when compared with the achievements of set theory and it brings into question the view that God predetermined an endless future. We therefore defend a new formal argument, based on the nature of time (...)
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  15. Elusive Propositions.Gabriel Uzquiano - 2021 - Journal of Philosophical Logic 50 (4):705-725.
    David Kaplan observed in Kaplan that the principle \\) cannot be verified at a world in a standard possible worlds model for a quantified bimodal propositional language. This raises a puzzle for certain interpretations of the operator Q: it seems that some proposition p is such that is not possible to query p, and p alone. On the other hand, Arthur Prior had observed in Prior that on pain of contradiction, ∀p is Q only if one true proposition is Q (...)
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  16. Thicket Density.Siddharth Bhaskar - 2021 - Journal of Symbolic Logic 86 (1):110-127.
    We define a new type of “shatter function” for set systems that satisfies a Sauer–Shelah type dichotomy, but whose polynomial-growth case is governed by Shelah’s two-rank instead of VC dimension. We identify the least exponent bounding the rate of growth of the shatter function, the quantity analogous to VC density, with Shelah’s $\omega $ -rank.
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  17. Separating Diagonal Stationary Reflection Principles.Gunter Fuchs & Chris Lambie-Hanson - 2021 - Journal of Symbolic Logic 86 (1):262-292.
    We introduce three families of diagonal reflection principles for matrices of stationary sets of ordinals. We analyze both their relationships among themselves and their relationships with other known principles of simultaneous stationary reflection, the strong reflection principle, and the existence of square sequences.
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  18. Recurrence and the Existence of Invariant Measures.Manuel J. Inselmann & Benjamin D. Miller - 2021 - Journal of Symbolic Logic 86 (1):60-76.
    We show that recurrence conditions do not yield invariant Borel probability measures in the descriptive set-theoretic milieu, in the strong sense that if a Borel action of a locally compact Polish group on a standard Borel space satisfies such a condition but does not have an orbit supporting an invariant Borel probability measure, then there is an invariant Borel set on which the action satisfies the condition but does not have an invariant Borel probability measure.
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  19. The Open and Clopen Ramsey Theorems in the Weihrauch Lattice.Alberto Marcone & Manlio Valenti - 2021 - Journal of Symbolic Logic 86 (1):316-351.
    We investigate the uniform computational content of the open and clopen Ramsey theorems in the Weihrauch lattice. While they are known to be equivalent to $\mathrm {ATR_0}$ from the point of view of reverse mathematics, there is not a canonical way to phrase them as multivalued functions. We identify eight different multivalued functions and study their degree from the point of view of Weihrauch, strong Weihrauch, and arithmetic Weihrauch reducibility. In particular one of our functions turns out to be strictly (...)
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  20. Modal Objectivity1.Clarke-Doane Justin - 2019 - Noûs:266-295.
    It is widely agreed that the intelligibility of modal metaphysics has been vindicated. Quine's arguments to the contrary supposedly confused analyticity with metaphysical necessity, and rigid with non-rigid designators.2 But even if modal metaphysics is intelligible, it could be misconceived. It could be that metaphysical necessity is not absolute necessity – the strictest real notion of necessity – and that no proposition of traditional metaphysical interest is necessary in every real sense. If there were nothing otherwise “uniquely metaphysically significant” about (...)
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  21. A Characterization of Σ11-Reflecting Ordinals.J. P. Aguilera - 2021 - Annals of Pure and Applied Logic 172 (10):103009.
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  22. A Descriptive Main Gap Theorem.Francesco Mangraviti & Luca Motto Ros - 2020 - Journal of Mathematical Logic 21 (1).
    Answering one of the main questions of [S.-D. Friedman, T. Hyttinen and V. Kulikov, Generalized descriptive set theory and classification theory, Mem. Amer. Math. Soc. 230 80, Chap. 7]...
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  23. On Negation for Non-classical Set Theories.S. Jockwich Martinez & G. Venturi - 2021 - Journal of Philosophical Logic 50 (3):549-570.
    We present a case study for the debate between the American and the Australian plans, analyzing a crucial aspect of negation: expressivity within a theory. We discuss the case of non-classical set theories, presenting three different negations and testing their expressivity within algebra-valued structures for ZF-like set theories. We end by proposing a minimal definitional account of negation, inspired by the algebraic framework discussed.
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  24. Criteria for Exact Saturation and Singular Compactness.Itay Kaplan, Nicholas Ramsey & Saharon Shelah - 2021 - Annals of Pure and Applied Logic 172 (9):102992.
    We introduce the class of unshreddable theories, which contains the simple and NIP theories, and prove that such theories have exactly saturated models in singular cardinals, satisfying certain set-theoretic hypotheses. We also give criteria for a theory to have singular compactness.
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  25. Interview With a Set Theorist.Deborah Kant & Mirna Džamonja - 2019 - In Stefania Centrone, Deborah Kant & Deniz Sarikaya (eds.), Reflections on the Foundations of Mathematics: Univalent Foundations, Set Theory and General Thoughts. Springer Verlag. pp. 3-26.
    The status of independent statements is the main problem in the philosophy of set theory. We address this problem by presenting the perspective of a practising set theorist. We thus give an authentic insight in the current state of thinking in set-theoretic practice, which is to a large extent determined by independence results. During several meetings, the second author asked the first author about the development of forcing, the use of new axioms and set-theoretic intuition on independence. Parts of these (...)
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  26. Tukey Order Among F Σ Ideals.Jialiang He, Michael Hrušák, Diego Rojas-Rebolledo & Sławomir Solecki - forthcoming - Journal of Symbolic Logic:1-15.
    We investigate the Tukey order in the class of Fσ ideals of subsets of ω. We show that no nontrivial Fσ ideal is Tukey below a Gδ ideal of compact sets. We introduce the notions of flat ideals and gradually flat ideals. We prove a dichotomy theorem for flat ideals isolating gradual flatness as the side of the dichotomy that is structurally good. We give diverse characterizations of gradual flatness among flat ideals using Tukey reductions and games. For example, we (...)
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  27. Closure Properties of Measurable Ultrapowers.Philipp Lücke & Sandra Müller - forthcoming - Journal of Symbolic Logic:1-22.
    We study closure properties of measurable ultrapowers with respect to Hamkin's notion of freshness and show that the extent of these properties highly depends on the combinatorial properties of the underlying model of set theory. In one direction, a result of Sakai shows that, by collapsing a strongly compact cardinal to become the double successor of a measurable cardinal, it is possible to obtain a model of set theory in which such ultrapowers possess the strongest possible closure properties. In the (...)
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  28. Level Theory, Part 1: Axiomatizing the Bare Idea of a Cumulative Hierarchy of Sets.Tim Button - forthcoming - Bulletin of Symbolic Logic:1-27.
    The following bare-bones story introduces the idea of a cumulative hierarchy of pure sets: 'Sets are arranged in stages. Every set is found at some stage. At any stage S: for any sets found before S, we find a set whose members are exactly those sets. We find nothing else at S.' Surprisingly, this story already guarantees that the sets are arranged in well-ordered levels, and suffices for quasi-categoricity. I show this by presenting Level Theory, a simplification of set theories (...)
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  29. Banach-Stone-Like Results for Combinatorial Banach Spaces.C. Brech & C. Piña - 2021 - Annals of Pure and Applied Logic 172 (8):102989.
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  30. Level Theory, Part 2: Axiomatizing the Bare Idea of a Potential Hierarchy.Tim Button - forthcoming - Bulletin of Symbolic Logic:1-25.
    Potentialists think that the concept of set is importantly modal. Using tensed language as an heuristic, the following bar-bones story introduces the idea of a potential hierarchy of sets: 'Always: for any sets that existed, there is a set whose members are exactly those sets; there are no other sets.' Surprisingly, this story already guarantees well-foundedness and persistence. Moreover, if we assume that time is linear, the ensuing modal set theory is almost definitionally equivalent with non-modal set theories; specifically, with (...)
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  31. Ontology, Set Theory, and the Paraphrase Challenge.Jared Warren - 2021 - Journal of Philosophical Logic 50 (6):1231-1248.
    In many ontological debates there is a familiar challenge. Consider a debate over X s. The “small” or anti-X side tries to show that they can paraphrase the pro-X or “big” side’s claims without any loss of expressive power. Typically though, when the big side adds whatever resources the small side used in their paraphrase, the symmetry breaks down. The big side plus small’s resources is a more expressively powerful and thus more theoretically fruitful theory. In this paper, I show (...)
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  32. Filling Cages. Reverse Mathematics and Combinatorial Principles.Marta Fiori Carones - 2020 - Bulletin of Symbolic Logic 26 (3-4):300-300.
    In the thesis some combinatorial statements are analysed from the reverse mathematics point of view. Reverse mathematics is a research program, which dates back to the Seventies, interested in finding the exact strength, measured in terms of set-existence axioms, of theorems from ordinary non set-theoretic mathematics. After a brief introduction to the subject, an on-line (incremental) algorithm to transitively reorient infinite pseudo-transitive oriented graphs is defined. This implies that a theorem of Ghouila-Houri is provable in RCA_0 and hence is computably (...)
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  33. An Exposition of the Compactness Of.Enrique Casanovas & Martin Ziegler - 2020 - Bulletin of Symbolic Logic 26 (3-4):212-218.
    We give an exposition of the compactness of L(QcfC), for any set C of regular cardinals.
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  34. Ontology and Arbitrariness.David Builes - forthcoming - Australasian Journal of Philosophy.
    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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  35. Around Rubin’s “Theories of Linear Order”.Predrag Tanović, Slavko Moconja & Dejan Ilić - 2020 - Journal of Symbolic Logic 85 (4):1403-1426.
    Let $\mathcal M=$ be a linearly ordered first-order structure and T its complete theory. We investigate conditions for T that could guarantee that $\mathcal M$ is not much more complex than some colored orders. Motivated by Rubin’s work [5], we label three conditions expressing properties of types of T and/or automorphisms of models of T. We prove several results which indicate the “geometric” simplicity of definable sets in models of theories satisfying these conditions. For example, we prove that the strongest (...)
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  36. A General Framework for a Second Philosophy Analysis of Set-Theoretic Methodology.Carolin Antos & Deborah Kant - manuscript
    Penelope Maddy’s Second Philosophy is one of the most well-known ap- proaches in recent philosophy of mathematics. She applies her second-philosophical method to analyze mathematical methodology by reconstructing historical cases in a setting of means-ends relations. However, outside of Maddy’s own work, this kind of methodological analysis has not yet been extensively used and analyzed. In the present work, we will make a first step in this direction. We develop a general framework that allows us to clarify the procedure and (...)
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  37. Independence Proofs in Non-Classical Set Theories.Sourav Tarafder & Giorgio Venturi - forthcoming - Review of Symbolic Logic:1-32.
  38. Simple-Like Independence Relations in Abstract Elementary Classes.Rami Grossberg & Marcos Mazari-Armida - 2021 - Annals of Pure and Applied Logic 172 (7):102971.
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  39. Bases for Functions Beyond the First Baire Class.Raphaël Carroy & Benjamin D. Miller - 2020 - Journal of Symbolic Logic 85 (3):1289-1303.
    We provide a finite basis for the class of Borel functions that are not in the first Baire class, as well as the class of Borel functions that are not $\sigma $ -continuous with closed witnesses.
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  40. Decomposing Functions of Baire Class on Polish Spaces.Longyun Ding, Takayuki Kihara, Brian Semmes & Jiafei Zhao - 2020 - Journal of Symbolic Logic 85 (3):960-971.
    We prove the Decomposability Conjecture for functions of Baire class $2$ from a Polish space to a separable metrizable space. This partially answers an important open problem in descriptive set theory.
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  41. Infinitary Generalizations of Deligne’s Completeness Theorem.Christian Espíndola - 2020 - Journal of Symbolic Logic 85 (3):1147-1162.
    Given a regular cardinal $\kappa $ such that $\kappa ^{<\kappa }=\kappa $, we study a class of toposes with enough points, the $\kappa $ -separable toposes. These are equivalent to sheaf toposes over a site with $\kappa $ -small limits that has at most $\kappa $ many objects and morphisms, the topology being generated by at most $\kappa $ many covering families, and that satisfy a further exactness property T. We prove that these toposes have enough $\kappa $ -points, that (...)
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  42. The Exact Strength of the Class Forcing Theorem.Victoria Gitman, Joel David Hamkins, Peter Holy, Philipp Schlicht & Kameryn J. Williams - 2020 - Journal of Symbolic Logic 85 (3):869-905.
    The class forcing theorem, which asserts that every class forcing notion ${\mathbb {P}}$ admits a forcing relation $\Vdash _{\mathbb {P}}$, that is, a relation satisfying the forcing relation recursion—it follows that statements true in the corresponding forcing extensions are forced and forced statements are true—is equivalent over Gödel–Bernays set theory $\text {GBC}$ to the principle of elementary transfinite recursion $\text {ETR}_{\text {Ord}}$ for class recursions of length $\text {Ord}$. It is also equivalent to the existence of truth predicates for the (...)
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  43. Introduction.Franklin D. Tall - 2021 - Annals of Pure and Applied Logic 172 (5):102902.
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  44. A Reconstruction of Steel’s Multiverse Project.Penelope Maddy & Toby Meadows - 2020 - Bulletin of Symbolic Logic 26 (2):118-169.
    This paper reconstructs Steel’s multiverse project in his ‘Gödel’s program’ (Steel [2014]), first by comparing it to those of Hamkins [2012] and Woodin [2011], then by detailed analysis what’s presented in Steel’s brief text. In particular, we reconstruct his notion of a ‘natural’ theory, describe his multiverse axioms and his translation function, and assess the resulting status of the Continuum Hypothesis. In the end, we reconceptualize the defect that Steel thinks CH might suffer from and isolate what it would take (...)
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  45. A Refinement of the Ramsey Hierarchy Via Indescribability.Brent Cody - 2020 - Journal of Symbolic Logic 85 (2):773-808.
    We study large cardinal properties associated with Ramseyness in which homogeneous sets are demanded to satisfy various transfinite degrees of indescribability. Sharpe and Welch [25], and independently Bagaria [1], extended the notion of $\Pi ^1_n$ -indescribability where $n<\omega $ to that of $\Pi ^1_\xi $ -indescribability where $\xi \geq \omega $. By iterating Feng’s Ramsey operator [12] on the various $\Pi ^1_\xi $ -indescribability ideals, we obtain new large cardinal hierarchies and corresponding nonlinear increasing hierarchies of normal ideals. We provide (...)
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  46. Exact Completion and Constructive Theories of Sets.Jacopo Emmenegger & Erik Palmgren - 2020 - Journal of Symbolic Logic 85 (2):563-584.
    In the present paper we use the theory of exact completions to study categorical properties of small setoids in Martin-Löf type theory and, more generally, of models of the Constructive Elementary Theory of the Category of Sets, in terms of properties of their subcategories of choice objects. Because of these intended applications, we deal with categories that lack equalisers and just have weak ones, but whose objects can be regarded as collections of global elements. In this context, we study the (...)
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  47. Sigma-Prikry Forcing II: Iteration Scheme.Alejandro Poveda, Assaf Rinot & Dima Sinapova - forthcoming - Journal of Mathematical Logic:2150019.
    In Part I of this series [5], 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 centers around singular cardinals of countable cofinality are [Formula: see text]-Prikry. We proved that given a [Formula: see text]-Prikry poset [Formula: see text] and a [Formula: see text]-name for a nonreflecting stationary set [Formula: see text], there exists a corresponding [Formula: see text]-Prikry poset that projects to [Formula: (...)
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  48. Existentially Closed Closure Algebras.Philip Scowcroft - 2020 - Notre Dame Journal of Formal Logic 61 (4):623-661.
    The study of existentially closed closure algebras begins with Lipparini’s 1982 paper. After presenting new nonelementary axioms for algebraically closed and existentially closed closure algebras and showing that these nonelementary classes are different, this paper shows that the classes of finitely generic and infinitely generic closure algebras are closed under finite products and bounded Boolean powers, extends part of Hausdorff’s theory of reducible sets to existentially closed closure algebras, and shows that finitely generic and infinitely generic closure algebras are elementarily (...)
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  49. Quantum Set Theory: Transfer Principle and De Morgan's Laws.Masanao Ozawa - 2021 - Annals of Pure and Applied Logic 172 (4):102938.
    In quantum logic, introduced by Birkhoff and von Neumann, De Morgan's Laws play an important role in the projection-valued truth value assignment of observational propositions in quantum mechanics. Takeuti's quantum set theory extends this assignment to all the set-theoretical statements on the universe of quantum sets. However, Takeuti's quantum set theory has a problem in that De Morgan's Laws do not hold between universal and existential bounded quantifiers. Here, we solve this problem by introducing a new truth value assignment for (...)
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  50. The Σ1-Definable Universal Finite Sequence.Joel David Hamkins & Kameryn J. Williams - forthcoming - Journal of Symbolic Logic:1-19.
1 — 50 / 2249