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  1. 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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  2. 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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  3. 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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  4. 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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  5. 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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  6. 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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  7. 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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  8. 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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  9. Wittgenstein on Cantor's Proof.Chrysoula Gitsoulis - 2019 - In Gabriele Mras, Paul Weingartner & Bernhard Ritter (eds.), Philosophy of Logic and Mathematics, Contributions to the 41st International Wittgenstein Symposium. Austrian Ludwig Wittgenstein Society. pp. 67-69.
    Cantor’s proof that the reals are uncountable forms a central pillar in the edifices of higher order recursion theory and set theory. It also has important applications in model theory, and in the foundations of topology and analysis. Due partly to these factors, and to the simplicity and elegance of the proof, it has come to be accepted as part of the ABC’s of mathematics. But even if as an Archimedean point it supports tomes of mathematical theory, there is a (...)
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  10. Why Believe Infinite Sets Exist?Andrei Mărăşoiu - 2018 - Axiomathes 28 (4):447-460.
    The axiom of infinity states that infinite sets exist. I will argue that this axiom lacks justification. I start by showing that the axiom is not self-evident, so it needs separate justification. Following Maddy’s :481–511, 1988) distinction, I argue that the axiom of infinity lacks both intrinsic and extrinsic justification. Crucial to my project is Skolem’s From Frege to Gödel: a source book in mathematical logic, 1879–1931, Cambridge, Harvard University Press, pp. 290–301, 1922) distinction between a theory of real sets, (...)
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  11. Actual and Potential Infinity.Øystein Linnebo & Stewart Shapiro - 2017 - Noûs 53 (1):160-191.
    The notion of potential infinity dominated in mathematical thinking about infinity from Aristotle until Cantor. The coherence and philosophical importance of the notion are defended. Particular attention is paid to the question of whether potential infinity is compatible with classical logic or requires a weaker logic, perhaps intuitionistic.
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  12. Is the Dream Solution of the Continuum Hypothesis Attainable?Joel David Hamkins - 2015 - Notre Dame Journal of Formal Logic 56 (1):135-145.
    The dream solution of the continuum hypothesis would be a solution by which we settle the continuum hypothesis on the basis of a newly discovered fundamental principle of set theory, a missing axiom, widely regarded as true. Such a dream solution would indeed be a solution, since we would all accept the new axiom along with its consequences. In this article, however, I argue that such a dream solution to $\mathrm {CH}$ is unattainable.
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  13. History and philosophy of infinity: Selected papers from the conference “Foundations of the Formal Sciences VIII” held at Corpus Christi College, Cambridge, England, 20–23 September 2013.Brendan P. Larvor, Benedikt Löwe & Dirk Schlimm - 2015 - Synthese 192 (8):2339-2344.
  14. 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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  15. What is the infinite?Øystein Linnebo - 2013 - The Philosophers' Magazine 61:42-47.
    The paper discusses some different conceptions of the infinity, from Aristotle to Georg Cantor (1845-1918) and beyond. The ancient distinction between actual and potential infinity is explained, along with some arguments against the possibility of actually infinite collections. These arguments were eventually rejected by most philosophers and mathematicians as a result of Cantor’s elegant and successful theory of actually infinite collections.
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  16. The Axiom of Infinity and Transformations j: V → V.Paul Corazza - 2010 - Bulletin of Symbolic Logic 16 (1):37-84.
    We suggest a new approach for addressing the problem of establishing an axiomatic foundation for large cardinals. An axiom asserting the existence of a large cardinal can naturally be viewed as a strong Axiom of Infinity. However, it has not been clear on the basis of our knowledge of ω itself, or of generally agreed upon intuitions about the true nature of the mathematical universe, what the right strengthening of the Axiom of Infinity is—which large cardinals ought to be derivable? (...)
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  17. 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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  18. Taming the Infinite. [REVIEW]Michael Potter - 1996 - British Journal for the Philosophy of Science 47 (4):609-619.
    A critique of Shaughan Lavine's attempt in /Understanding the Infinite/ to reduce talk about the infinite to finitely comprehensible terms.
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  19. Properties, abstracts, and the axiom of infinity.Herbert Hochberg - 1977 - Journal of Philosophical Logic 6 (1):193 - 207.
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  20. Reduction of arithmetic to logic based on the theory of types without the axiom of infinity and the typical ambiguity of arithmetical constants.Ludwik Borkowski - 1958 - Studia Logica 8 (1):283 - 297.
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  21. On ω-inconsistency and a so-called axiom of infinity.W. V. Quine - 1953 - Journal of Symbolic Logic 18 (2):119-124.
  22. The axiom of infinity in Quine's new foundations.J. Barkley Rosser - 1952 - Journal of Symbolic Logic 17 (4):238-242.
    We use NF to designate the system known as Quine's New Foundations, and NF + AF to designate the same system with a suitable axiom of infinity adjoined. We use ML to designate the revised system appearing in the third printing of Quine's “Mathematical Logic”. This system ML is just the systemPproposed by Wang in [4], and essentially includes NF as a part.The pripcipal results of the present paper are:A. In NF the axiom of infinity is equivalent to the definability (...)
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  23. Hilbert's machine and the axiom of infinity.Antonio Leon - 2006
    Hilbert's machine is a supertask machine inspired by Hilbert's Hotel whose functioning leads to a contradiction that compromises the Axiom of Infinity.
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