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  1. Russell's-Paradox-Intercepting Corollary to the Axiom of Extensionality.Morteza Shahram - manuscript
    Object x being a member of itself or not and x being a member of R or not constitute two vastly different concepts. This paper attempts to locate the reflection of such an utter difference within the formal structure of the axiom of extensionality. -/- .
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  2. Executing Gödel's Programme in Set Theory.Neil Barton - 2017 - Dissertation, Birkbeck, University of London
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  3. Nota: ¿CUÁL ES EL CARDINAL DEL CONJUNTO DE LOS NÚMEROS REALES?Franklin Galindo - manuscript
    ¿Qué ha pasado con el problema del cardinal del continuo después de Gödel (1938) y Cohen (1964)? Intentos de responder esta pregunta pueden encontrarse en los artículos de José Alfredo Amor (1946-2011), "El Problema del continuo después de Cohen (1964-2004)", de Carlos Di Prisco , "Are we closer to a solution of the continuum problem", y de Joan Bagaria, "Natural axioms of set and the continuum problem" , que se pueden encontrar en la biblioteca digital de mi blog de Lógica (...)
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  4. Chance and the Continuum Hypothesis.Daniel Hoek - 2020 - Philosophy and Phenomenological Research 103 (3):639-60.
    This paper presents and defends an argument that the continuum hypothesis is false, based on considerations about objective chance and an old theorem due to Banach and Kuratowski. More specifically, I argue that the probabilistic inductive methods standardly used in science presuppose that every proposition about the outcome of a chancy process has a certain chance between 0 and 1. I also argue in favour of the standard view that chances are countably additive. Since it is possible to randomly pick (...)
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  5. (2 other versions)Hyperintensional Ω-Logic.David Elohim - 2019 - In Matteo Vincenzo D'Alfonso & Don Berkich (eds.), On the Cognitive, Ethical, and Scientific Dimensions of Artificial Intelligence. Springer Verlag. pp. 65-82.
    This paper examines the philosophical significance of the consequence relation defined in the $\Omega$-logic for set-theoretic languages. I argue that, as with second-order logic, the hyperintensional profile of validity in $\Omega$-Logic enables the property to be epistemically tractable. Because of the duality between coalgebras and algebras, Boolean-valued models of set theory can be interpreted as coalgebras. In Section \textbf{2}, I demonstrate how the hyperintensional profile of $\Omega$-logical validity can be countenanced within a coalgebraic logic. Finally, in Section \textbf{3}, the philosophical (...)
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  6. Modal set theory.Christopher Menzel - 2018 - In Otávio Bueno & Scott A. Shalkowski (eds.), The Routledge Handbook of Modality. New York: Routledge.
    This article presents an overview of the basic philosophical motivations for, and some recent work in, modal set theory.
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  7. Epistemology of Logic - Logic-Dialectic or Theory of the Knowledge.Epameinondas Xenopoulos - 1998 - Kefalonia,GREECE: KATERINA XENOPOULOU.
    1994.Επιστημολογία της Λογικής. Συγγραφέας Επαμεινώνδας Ξενόπουλος Μοναδική μελέτη και προσέγγιση της θεωρίας της γνώσης, για την παγκόσμια βιβλιογραφία, της διαλεκτικής πορείας της σκέψης από την λογική πλευρά της και της μελλοντικής μορφής που θα πάρουν οι διαλεκτικές δομές της, στην αδιαίρετη ενότητα γνωσιοθεωρίας, λογικής και διαλεκτικής, με την «μέθοδο του διαλεκτικού υλισμού». Έργο βαρύ με θέμα εξαιρετικά δύσκολο διακατέχεται από πρωτοτυπία και ζωντάνια που γοητεύει τον κάθε ανήσυχο στοχαστή από τις πρώτες γραμμές.
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  8. The Axiom of choice in Quine's New Foundations for Mathematical Logic.Ernst P. Specker - 1954 - Journal of Symbolic Logic 19 (2):127-128.
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  9. The axiom of infinity.Bertrand Russell - 1903 - Hibbert Journal 2:809-812.
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  10. The axiom of infinity: A new presupposition of thought.Cassius Jackson Keyser - 1903 - Hibbert Journal 2:532-552.
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  11. The Axiom of Infinity.Cassius Jackson Keyser - 1904 - Hibbert Journal 3:380-383.
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The Axiom of Choice
  1. Hume’s Principle, Bad Company, and the Axiom of Choice.Sam Roberts & Stewart Shapiro - 2023 - Review of Symbolic Logic 16 (4):1158-1176.
    One prominent criticism of the abstractionist program is the so-called Bad Company objection. The complaint is that abstraction principles cannot in general be a legitimate way to introduce mathematical theories, since some of them are inconsistent. The most notorious example, of course, is Frege’s Basic Law V. A common response to the objection suggests that an abstraction principle can be used to legitimately introduce a mathematical theory precisely when it is stable: when it can be made true on all sufficiently (...)
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  2. The Banach-Tarski Paradox.Ulrich Meyer - forthcoming - Logique Et Analyse.
    Emile Borel regards the Banach-Tarski Paradox as a reductio ad absurdum of the Axiom of Choice. Peter Forrest instead blames the assumption that physical space has a similar structure as the real numbers. This paper argues that Banach and Tarski's result is not paradoxical and that it merely illustrates a surprising feature of the continuum: dividing a spatial region into disjoint pieces need not preserve volume.
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  3. Sufficient Reason & The Axiom of Choice, an Ontological Proof for One Unique Transcendental God for Every Possible World.Assem Hamdy - manuscript
    Chains of causes appear when the existence of God is discussed. It is claimed by some that these chains must be finite and terminated by God. But these chains seem endless through our knowledge search. This endlessness for the physical reasons for any world event expresses the greatness and complexity of God’s creation and so the transcendence of God. So, only we can put our hands on physical reasons in an endless forage for knowledge. Yet, the endlessness of the physical (...)
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  4. Hilbert Mathematics versus Gödel Mathematics. III. Hilbert Mathematics by Itself, and Gödel Mathematics versus the Physical World within It: both as Its Particular Cases.Vasil Penchev - 2023 - Philosophy of Science eJournal (Elsevier: SSRN) 16 (47):1-46.
    The paper discusses Hilbert mathematics, a kind of Pythagorean mathematics, to which the physical world is a particular case. The parameter of the “distance between finiteness and infinity” is crucial. Any nonzero finite value of it features the particular case in the frameworks of Hilbert mathematics where the physical world appears “ex nihilo” by virtue of an only mathematical necessity or quantum information conservation physically. One does not need the mythical Big Bang which serves to concentrate all the violations of (...)
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  5. Gödel mathematics versus Hilbert mathematics. I. The Gödel incompleteness (1931) statement: axiom or theorem?Vasil Penchev - 2022 - Logic and Philosophy of Mathematics eJournal (Elsevier: SSRN) 14 (9):1-56.
    The present first part about the eventual completeness of mathematics (called “Hilbert mathematics”) is concentrated on the Gödel incompleteness (1931) statement: if it is an axiom rather than a theorem inferable from the axioms of (Peano) arithmetic, (ZFC) set theory, and propositional logic, this would pioneer the pathway to Hilbert mathematics. One of the main arguments that it is an axiom consists in the direct contradiction of the axiom of induction in arithmetic and the axiom of infinity in set theory. (...)
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  6. El Axioma de elección en el quehacer matemático contemporáneo.Franklin Galindo & Randy Alzate - 2022 - Aitías 2 (3):49-126.
    Para matemáticos interesados en problemas de fundamentos, lógico-matemáticos y filósofos de la matemática, el axioma de elección es centro obligado de reflexión, pues ha sido considerado esencial en el debate dentro de las posiciones consideradas clásicas en filosofía de la matemática (intuicionismo, formalismo, logicismo, platonismo), pero también ha tenido una presencia fundamental para el desarrollo de la matemática y metamatemática contemporánea. Desde una posición que privilegia el quehacer matemático, nos proponemos mostrar los aportes que ha tenido el axioma en varias (...)
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  7. Gentzen’s “cut rule” and quantum measurement in terms of Hilbert arithmetic. Metaphor and understanding modeled formally.Vasil Penchev - 2022 - Logic and Philosophy of Mathematics eJournal 14 (14):1-37.
    Hilbert arithmetic in a wide sense, including Hilbert arithmetic in a narrow sense consisting by two dual and anti-isometric Peano arithmetics, on the one hand, and the qubit Hilbert space (originating for the standard separable complex Hilbert space of quantum mechanics), on the other hand, allows for an arithmetic version of Gentzen’s cut elimination and quantum measurement to be described uniformy as two processes occurring accordingly in those two branches. A philosophical reflection also justifying that unity by quantum neo-Pythagoreanism links (...)
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  8. Un problema abierto de independencia en la teoría de conjuntos relacionado con ultrafiltros no principales sobre el conjunto de los números naturales N, y con Propiedades Ramsey.Franklin Galindo - manuscript
    En el ámbito de la lógica matemática existe un problema sobre la relación lógica entre dos versiones débiles del Axioma de elección (AE) que no se ha podido resolver desde el año 2000 (aproximadamente). Tales versiones están relacionadas con ultrafiltros no principales y con Propiedades Ramsey (Bernstein, Polarizada, Subretículo, Ramsey, Ordinales flotantes, etc). La primera versión débil del AE es la siguiente (A): “Existen ultrafiltros no principales sobre el conjunto de los números naturales (ℕ)”. Y la segunda versión débil del (...)
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  9. Constructibilidad relativizada y el Axioma de elección.Franklin Galindo & Carlos Di Prisco - 2010 - Mixba'al. Revista Metropolitana de Matemáticas 1 (1):23-40.
    El objetivo de este trabajo es presentar en un solo cuerpo tres maneras de relativizar (o generalizar) el concepto de conjunto constructible de Gödel que no suelen aparecer juntas en la literatura especializada y que son importantes en la Teoría de Conjuntos, por ejemplo para resolver problemas de consistencia o independencia. Presentamos algunos modelos resultantes de las diferentes formas de relativizar el concepto de constructibilidad, sus propiedades básicas y algunas formas débiles del Axioma de Elección válidas o no válidas en (...)
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  10. Perfect set properties in models of ZF.Franklin Galindo & Carlos Di Prisco - 2010 - Fundamenta Mathematicae 208 (208):249-262.
    We study several perfect set properties of the Baire space which follow from the Ramsey property ω→(ω) ω . In particular we present some independence results which complete the picture of how these perfect set properties relate to each other.
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  11. Álgebras booleanas, órdenes parciales y axioma de elección.Franklin Galindo - 2017 - Divulgaciones Matematicas 18 ( 1):34-54.
    El objetivo de este artículo es presentar una demostración de un teorema clásico sobre álgebras booleanas y ordenes parciales de relevancia actual en teoría de conjuntos, como por ejemplo, para aplicaciones del método de construcción de modelos llamado “forcing” (con álgebras booleanas completas o con órdenes parciales). El teorema que se prueba es el siguiente: “Todo orden parcial se puede extender a una única álgebra booleana completa (salvo isomorfismo)”. Donde extender significa “sumergir densamente”. Tal demostración se realiza utilizando cortaduras de (...)
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  12. The Gödel Incompleteness Theorems (1931) by the Axiom of Choice.Vasil Penchev - 2020 - Econometrics: Mathematical Methods and Programming eJournal (Elsevier: SSRN) 13 (39):1-4.
    Those incompleteness theorems mean the relation of (Peano) arithmetic and (ZFC) set theory, or philosophically, the relation of arithmetical finiteness and actual infinity. The same is managed in the framework of set theory by the axiom of choice (respectively, by the equivalent well-ordering "theorem'). One may discuss that incompleteness form the viewpoint of set theory by the axiom of choice rather than the usual viewpoint meant in the proof of theorems. The logical corollaries from that "nonstandard" viewpoint the relation of (...)
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  13. Quantum Invariance.Vasil Penchev - 2020 - Epistemology eJournal (Elsevier: SSRN) 13 (22):1-6.
    Quantum invariance designates the relation of any quantum coherent state to the corresponding statistical ensemble of measured results. The adequate generalization of ‘measurement’ is discussed to involve the discrepancy, due to the fundamental Planck constant, between any quantum coherent state and its statistical representation as a statistical ensemble after measurement. A set-theory corollary is the curious invariance to the axiom of choice: Any coherent state excludes any well-ordering and thus excludes also the axiom of choice. It should be equated to (...)
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  14. High-Order Metaphysics as High-Order Abstractions and Choice in Set Theory.Vasil Penchev - 2020 - Epistemology eJournal (Elsevier: SSRN) 13 (21):1-3.
    The link between the high-order metaphysics and abstractions, on the one hand, and choice in the foundation of set theory, on the other hand, can distinguish unambiguously the “good” principles of abstraction from the “bad” ones and thus resolve the “bad company problem” as to set theory. Thus it implies correspondingly a more precise definition of the relation between the axiom of choice and “all company” of axioms in set theory concerning directly or indirectly abstraction: the principle of abstraction, axiom (...)
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  15. Choice, Infinity, and Negation: Both Set-Theory and Quantum-Information Viewpoints to Negation.Vasil Penchev - 2020 - Logic and Philosophy of Mathematics eJournal 12 (14):1-3.
    The concepts of choice, negation, and infinity are considered jointly. The link is the quantity of information interpreted as the quantity of choices measured in units of elementary choice: a bit is an elementary choice between two equally probable alternatives. “Negation” supposes a choice between it and confirmation. Thus quantity of information can be also interpreted as quantity of negations. The disjunctive choice between confirmation and negation as to infinity can be chosen or not in turn: This corresponds to set-theory (...)
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  16. Russell’s method of analysis and the axioms of mathematics.Lydia Patton - 2017 - In Sandra Lapointe & Christopher Pincock (eds.), Innovations in the History of Analytical Philosophy. London, United Kingdom: Palgrave-Macmillan. pp. 105-126.
    In the early 1900s, Russell began to recognize that he, and many other mathematicians, had been using assertions like the Axiom of Choice implicitly, and without explicitly proving them. In working with the Axioms of Choice, Infinity, and Reducibility, and his and Whitehead’s Multiplicative Axiom, Russell came to take the position that some axioms are necessary to recovering certain results of mathematics, but may not be proven to be true absolutely. The essay traces historical roots of, and motivations for, Russell’s (...)
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The Axiom of Constructibility
  1. Constructibilidad relativizada y el Axioma de elección.Franklin Galindo & Carlos Di Prisco - 2010 - Mixba'al. Revista Metropolitana de Matemáticas 1 (1):23-40.
    El objetivo de este trabajo es presentar en un solo cuerpo tres maneras de relativizar (o generalizar) el concepto de conjunto constructible de Gödel que no suelen aparecer juntas en la literatura especializada y que son importantes en la Teoría de Conjuntos, por ejemplo para resolver problemas de consistencia o independencia. Presentamos algunos modelos resultantes de las diferentes formas de relativizar el concepto de constructibilidad, sus propiedades básicas y algunas formas débiles del Axioma de Elección válidas o no válidas en (...)
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  2. The Metamathematics of Putnam’s Model-Theoretic Arguments.Tim Button - 2011 - Erkenntnis 74 (3):321-349.
    Putnam famously attempted to use model theory to draw metaphysical conclusions. His Skolemisation argument sought to show metaphysical realists that their favourite theories have countable models. His permutation argument sought to show that they have permuted models. His constructivisation argument sought to show that any empirical evidence is compatible with the Axiom of Constructibility. Here, I examine the metamathematics of all three model-theoretic arguments, and I argue against Bays (2001, 2007) that Putnam is largely immune to metamathematical challenges.
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The Axiom of Infinity
  1. (1 other version)Forever Finite: The Case Against Infinity (Expanded Edition).Kip K. Sewell - 2023 - Alexandria, VA: Rond Books.
    EXPANDED EDITION (eBook): -/- Infinity Is Not What It Seems...Infinity is commonly assumed to be a logical concept, reliable for conducting mathematics, describing the Universe, and understanding the divine. Most of us are educated to take for granted that there exist infinite sets of numbers, that lines contain an infinite number of points, that space is infinite in expanse, that time has an infinite succession of events, that possibilities are infinite in quantity, and over half of the world’s population believes (...)
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  2. Nothing Infinite: A Summary of Forever Finite.Kip Sewell - 2023 - Rond Media Library.
    In 'Forever Finite: The Case Against Infinity' (Rond Books, 2023), the author argues that, despite its cultural popularity, infinity is not a logical concept and consequently cannot be a property of anything that exists in the real world. This article summarizes the main points in 'Forever Finite', including its overview of what debunking infinity entails for conceptual thought in philosophy, mathematics, science, cosmology, and theology.
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  3. Hilbert Mathematics Versus Gödel Mathematics. IV. The New Approach of Hilbert Mathematics Easily Resolving the Most Difficult Problems of Gödel Mathematics.Vasil Penchev - 2023 - Philosophy of Science eJournal (Elsevier: SSRN) 16 (75):1-52.
    The paper continues the consideration of Hilbert mathematics to mathematics itself as an additional “dimension” allowing for the most difficult and fundamental problems to be attacked in a new general and universal way shareable between all of them. That dimension consists in the parameter of the “distance between finiteness and infinity”, particularly able to interpret standard mathematics as a particular case, the basis of which are arithmetic, set theory and propositional logic: that is as a special “flat” case of Hilbert (...)
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  4. Hilbert Mathematics versus Gödel Mathematics. III. Hilbert Mathematics by Itself, and Gödel Mathematics versus the Physical World within It: both as Its Particular Cases.Vasil Penchev - 2023 - Philosophy of Science eJournal (Elsevier: SSRN) 16 (47):1-46.
    The paper discusses Hilbert mathematics, a kind of Pythagorean mathematics, to which the physical world is a particular case. The parameter of the “distance between finiteness and infinity” is crucial. Any nonzero finite value of it features the particular case in the frameworks of Hilbert mathematics where the physical world appears “ex nihilo” by virtue of an only mathematical necessity or quantum information conservation physically. One does not need the mythical Big Bang which serves to concentrate all the violations of (...)
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  5. Gentzen’s “cut rule” and quantum measurement in terms of Hilbert arithmetic. Metaphor and understanding modeled formally.Vasil Penchev - 2022 - Logic and Philosophy of Mathematics eJournal 14 (14):1-37.
    Hilbert arithmetic in a wide sense, including Hilbert arithmetic in a narrow sense consisting by two dual and anti-isometric Peano arithmetics, on the one hand, and the qubit Hilbert space (originating for the standard separable complex Hilbert space of quantum mechanics), on the other hand, allows for an arithmetic version of Gentzen’s cut elimination and quantum measurement to be described uniformy as two processes occurring accordingly in those two branches. A philosophical reflection also justifying that unity by quantum neo-Pythagoreanism links (...)
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  6. Quantum Invariance.Vasil Penchev - 2020 - Epistemology eJournal (Elsevier: SSRN) 13 (22):1-6.
    Quantum invariance designates the relation of any quantum coherent state to the corresponding statistical ensemble of measured results. The adequate generalization of ‘measurement’ is discussed to involve the discrepancy, due to the fundamental Planck constant, between any quantum coherent state and its statistical representation as a statistical ensemble after measurement. A set-theory corollary is the curious invariance to the axiom of choice: Any coherent state excludes any well-ordering and thus excludes also the axiom of choice. It should be equated to (...)
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  7. High-Order Metaphysics as High-Order Abstractions and Choice in Set Theory.Vasil Penchev - 2020 - Epistemology eJournal (Elsevier: SSRN) 13 (21):1-3.
    The link between the high-order metaphysics and abstractions, on the one hand, and choice in the foundation of set theory, on the other hand, can distinguish unambiguously the “good” principles of abstraction from the “bad” ones and thus resolve the “bad company problem” as to set theory. Thus it implies correspondingly a more precise definition of the relation between the axiom of choice and “all company” of axioms in set theory concerning directly or indirectly abstraction: the principle of abstraction, axiom (...)
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  8. Dialogue on Alternating Consciousness: From Perception to Infinities and Back to Free Will.Claus Janew - 2014 - Journal of Consciousness Exploration and Research 5 (4):351-391.
    Can we trace back consciousness, reality, awareness, and free will to a single basic structure without giving up any of them? Can the universe exist in both real and individual ways without being composed of both? This dialogue founds consciousness and freedom of choice on the basis of a new reality concept that also includes the infinite as far as we understand it. Just the simplest distinction contains consciousness. It is not static, but a constant alternation of perspectives. From its (...)
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  9. God and Science are Infinite.John Linus O'Sullivan - forthcoming - AuthorsDen.
    Abstract: Standing half wave particles at light speed twice in expansion-contraction comprise a static universe where two transverse fields 90° out of phase are the square of distance from each other. The universe has a static concept of time since the infinite universe is a static universe without a beginning or end. The square of distance is a point of reversal in expansion-contraction between the fields as a means to conserve energy. Photons on expansion in the electric field create matter (...)
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  10. Infinitely Complex Machines.Eric Steinhart - 2007 - In Intelligent Computing Everywhere. Springer. pp. 25-43.
    Infinite machines (IMs) can do supertasks. A supertask is an infinite series of operations done in some finite time. Whether or not our universe contains any IMs, they are worthy of study as upper bounds on finite machines. We introduce IMs and describe some of their physical and psychological aspects. An accelerating Turing machine (an ATM) is a Turing machine that performs every next operation twice as fast. It can carry out infinitely many operations in finite time. Many ATMs can (...)
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New Axioms in Set Theory
  1. (2 other versions)Hyperintensional Ω-Logic.David Elohim - 2019 - In Matteo Vincenzo D'Alfonso & Don Berkich (eds.), On the Cognitive, Ethical, and Scientific Dimensions of Artificial Intelligence. Springer Verlag.
    This essay examines the philosophical significance of $\Omega$-logic in Zermelo-Fraenkel set theory with choice (ZFC). The categorical duality between coalgebra and algebra permits Boolean-valued algebraic models of ZFC to be interpreted as coalgebras. The hyperintensional profile of $\Omega$-logical validity can then be countenanced within a coalgebraic logic. I argue that the philosophical significance of the foregoing is two-fold. First, because the epistemic and modal and hyperintensional profiles of $\Omega$-logical validity correspond to those of second-order logical consequence, $\Omega$-logical validity is genuinely (...)
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  2. Fichte’s Formal Logic.Jens Lemanski & Andrew Schumann - 2023 - Synthese 202 (1):1-27.
    Fichte’s Foundations of the Entire Wissenschaftslehre 1794 is one of the most fundamental books in classical German philosophy. The use of laws of thought to establish foundational principles of transcendental philosophy was groundbreaking in the late eighteenth and early nineteenth century and is still crucial for many areas of theoretical philosophy and logic in general today. Nevertheless, contemporaries have already noted that Fichte’s derivation of foundational principles from the law of identity is problematic, since Fichte lacked the tools to correctly (...)
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  3. (1 other version)Iterative Conceptions of Set.Neil Barton - manuscript
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  4. Countabilism and Maximality Principles.Neil Barton & Sy-David Friedman - manuscript
    It is standard in set theory to assume that Cantor's Theorem establishes that the continuum is an uncountable set. A challenge for this position comes from the observation that through forcing one can collapse any cardinal to the countable and that the continuum can be made arbitrarily large. In this paper, we present a different take on the relationship between Cantor's Theorem and extensions of universes, arguing that they can be seen as showing that every set is countable and that (...)
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  5. (1 other version)Universism and extensions of V.Carolin Antos, Neil Barton & Sy-David Friedman - 2021 - Review of Symbolic Logic 14 (1):112-154.
    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 model-theoretic constructions that add sets to models are cited as evidence in favour of the latter. This paper informs this debate by developing a way for a Universist to interpret talk that (...)
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  6. (1 other version)Are Large Cardinal Axioms Restrictive?Neil Barton - manuscript
    The independence phenomenon in set theory, while pervasive, can be partially addressed through the use of large cardinal axioms. A commonly assumed idea is that large cardinal axioms are species of maximality principles. In this paper, I argue that whether or not large cardinal axioms count as maximality principles depends on prior commitments concerning the richness of the subset forming operation. In particular I argue that there is a conception of maximality through absoluteness, on which large cardinal axioms are restrictive. (...)
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  7. Maximality and ontology: how axiom content varies across philosophical frameworks.Sy-David Friedman & Neil Barton - 2017 - Synthese 197 (2):623-649.
    Discussion of new axioms for set theory has often focused on conceptions of maximality, and how these might relate to the iterative conception of set. This paper provides critical appraisal of how certain maximality axioms behave on different conceptions of ontology concerning the iterative conception. In particular, we argue that forms of multiversism (the view that any universe of a certain kind can be extended) and actualism (the view that there are universes that cannot be extended in particular ways) face (...)
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  8. (2 other versions)The Search for New Axioms in the Hyperuniverse Programme.Claudio Ternullo & Sy-David Friedman - 2016 - In Francesca Boccuni & Andrea Sereni (eds.), Objectivity, Realism, and Proof. FilMat Studies in the Philosophy of Mathematics. Cham, Switzerland: Springer International Publishing. pp. 165-188.
    The Hyperuniverse Programme, introduced in Arrigoni and Friedman (2013), fosters the search for new set-theoretic axioms. In this paper, we present the procedure envisaged by the programme to find new axioms and the conceptual framework behind it. The procedure comes in several steps. Intrinsically motivated axioms are those statements which are suggested by the standard concept of set, i.e. the `maximal iterative concept', and the programme identi fies higher-order statements motivated by the maximal iterative concept. The satisfaction of these statements (...)
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  9. Ipotesi del Continuo.Claudio Ternullo - 2017 - Aphex 16.
    L’Ipotesi del Continuo, formulata da Cantor nel 1878, è una delle congetture più note della teoria degli insiemi. Il Problema del Continuo, che ad essa è collegato, fu collocato da Hilbert, nel 1900, fra i principali problemi insoluti della matematica. A seguito della dimostrazione di indipendenza dell’Ipotesi del Continuo dagli assiomi della teoria degli insiemi, lo status attuale del problema è controverso. In anni più recenti, la ricerca di una soluzione del Problema del Continuo è stata anche una delle ragioni (...)
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  10. Richness and Reflection.Neil Barton - 2016 - Philosophia Mathematica 24 (3):330-359.
    A pervasive thought in contemporary philosophy of mathematics is that in order to justify reflection principles, one must hold universism: the view that there is a single universe of pure sets. I challenge this kind of reasoning by contrasting universism with a Zermelian form of multiversism. I argue that if extant justifications of reflection principles using notions of richness are acceptable for the universist, then the Zermelian can use similar justifications. However, I note that for some forms of richness argument, (...)
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  11. Note on Absolute Provability and Cantorian Comprehension.Holger A. Leuz - manuscript
    We will explicate Cantor’s principle of set existence using the Gödelian intensional notion of absolute provability and John Burgess’ plural logical concept of set formation. From this Cantorian Comprehension principle we will derive a conditional result about the question whether there are any absolutely unprovable mathematical truths. Finally, we will discuss the philosophical significance of the conditional result.
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