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  1. Canonical Maps.Jean-Pierre Marquis - 2018 - In Elaine Landry (ed.), Categories for the Working Philosophers. Oxford, UK: pp. 90-112.
    Categorical foundations and set-theoretical foundations are sometimes presented as alternative foundational schemes. So far, the literature has mostly focused on the weaknesses of the categorical foundations. We want here to concentrate on what we take to be one of its strengths: the explicit identification of so-called canonical maps and their role in mathematics. Canonical maps play a central role in contemporary mathematics and although some are easily defined by set-theoretical tools, they all appear systematically in a categorical framework. The key (...)
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  2. Dummett on Indefinite Extensibility.Øystein Linnebo - 2018 - Philosophical Issues 28 (1):196-220.
    Dummett’s notion of indefinite extensibility is influential but obscure. The notion figures centrally in an alternative Dummettian argument for intuitionistic logic and anti-realism, distinct from his more famous, meaning-theoretic arguments to the same effect. Drawing on ideas from Dummett, a precise analysis of indefinite extensibility is proposed. This analysis is used to reconstruct the poorly understood alternative argument. The plausibility of the resulting argument is assessed.
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  3. A Naturalistic Justification of the Generic Multiverse with a Core.Matteo de Ceglie - 2018 - In Proceedings of the 41st Internation Wittgenstein Symposium. 2880 Kirchberg am Wechsel, Austria: pp. 34-36.
    In this paper, I argue that a naturalist approach in philosophy of mathematics justifies a pluralist conception of set theory. For the pluralist, there is not a Single Universe, but there is rather a Multiverse, composed by a plurality of universes generated by various set theories. In order to justify a pluralistic approach to sets, I apply the two naturalistic principles developed by Penelope Maddy (cfr. Maddy (1997)), UNIFY and MAXIMIZE, and analyze through them the potential of the set theoretic (...)
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  4. Constructive Set Theory with Operations.Andrea Cantini & Laura Crosilla - 2008 - In Logic Colloquium 2004.
    We present an extension of constructive Zermelo{Fraenkel set theory [2]. Constructive sets are endowed with an applicative structure, which allows us to express several set theoretic constructs uniformly and explicitly. From the proof theoretic point of view, the addition is shown to be conservative. In particular, we single out a theory of constructive sets with operations which has the same strength as Peano arithmetic.
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  5. Elementary Constructive Operational Set Theory.Andrea Cantini & Laura Crosilla - 2010 - In Ways of Proof Theory.
    We introduce an operational set theory in the style of [5] and [16]. The theory we develop here is a theory of constructive sets and operations. One motivation behind constructive operational set theory is to merge a constructive notion of set ([1], [2]) with some aspects which are typical of explicit mathematics [14]. In particular, one has non-extensional operations (or rules) alongside extensional constructive sets. Operations are in general partial and a limited form of self{application is permitted. The system we (...)
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  6. Note on the Significance of the New Logic.Frederique Janssen-Lauret - 2018 - The Reasoner 6 (12):47-48.
    Brief note explaining the content, importance, and historical context of my joint translation of Quine's The Significance of the New Logic with my single-authored historical-philosophical essay 'Willard Van Orman Quine's Philosophical Development in the 1930s and 1940s'.
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  7. The Significance of the New Logic.Walter Carnielli, Frederique Janssen-Lauret & William Pickering (eds.) - 2018 - Cambridge: Cambridge University Press.
    W. V. Quine was one of the most influential figures of twentieth-century American analytic philosophy. Although he wrote predominantly in English, in Brazil in 1942 he gave a series of lectures on logic and its philosophy in Portuguese, subsequently published as the book O Sentido da Nova Lógica. The book has never before been fully translated into English, and this volume is the first to make its content accessible to Anglophone philosophers. Quine would go on to develop revolutionary ideas about (...)
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  8. The Foundations of Mathematics in the Theory of Sets.Roy T. Cook - 2003 - British Journal for the Philosophy of Science 54 (2):347-352.
  9. The Ins and Outs of Frege's Way Out.Gregory Landini - 2006 - Philosophia Mathematica 14 (1):1-25.
    Confronted with Russell's Paradox, Frege wrote an appendix to volume II of his _Grundgesetze der Arithmetik_. In it he offered a revision to Basic Law V, and proclaimed with confidence that the major theorems for arithmetic are recoverable. This paper shows that Frege's revised system has been seriously undermined by interpretations that transcribe his system into a predicate logic that is inattentive to important details of his concept-script. By examining the revised system as a concept-script, we see how Frege imagined (...)
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  10. 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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  11. The Foundations of Mathematics and Other Logical Essays.Frank Plumpton Ramsey - 1925 - Routledge & Kegan Paul.
  12. Russell Bektrand, Logical Positivism. Polemic , No. 1 , Pp. 6–13.Max Black - 1947 - Journal of Symbolic Logic 12 (1):24-24.
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  13. The Strength of Abstraction with Predicative Comprehension.Sean Walsh - 2016 - Bulletin of Symbolic Logic 22 (1):105–120.
    Frege's theorem says that second-order Peano arithmetic is interpretable in Hume's Principle and full impredicative comprehension. Hume's Principle is one example of an abstraction principle, while another paradigmatic example is Basic Law V from Frege's Grundgesetze. In this paper we study the strength of abstraction principles in the presence of predicative restrictions on the comprehension schema, and in particular we study a predicative Fregean theory which contains all the abstraction principles whose underlying equivalence relations can be proven to be equivalence (...)
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  14. The Potential Hierarchy of Sets.Øystein Linnebo - 2013 - Review of Symbolic Logic 6 (2):205-228.
    Some reasons to regard the cumulative hierarchy of sets as potential rather than actual are discussed. Motivated by this, a modal set theory is developed which encapsulates this potentialist conception. The resulting theory is equi-interpretable with Zermelo Fraenkel set theory but sheds new light on the set-theoretic paradoxes and the foundations of set theory.
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  15. The Barber, Russell's Paradox, Catch-22, God, Contradiction, and More.Laurence Goldstein - 2004 - In Graham Priest, J. C. Beall & Bradley Armour-Garb (eds.), The Law of Non-Contradiction. Clarendon Press. pp. 295--313.
    outrageous remarks about contradictions. Perhaps the most striking remark he makes is that they are not false. This claim first appears in his early notebooks (Wittgenstein 1960, p.108). In the Tractatus, Wittgenstein argued that contradictions (like tautologies) are not statements (Sätze) and hence are not false (or true). This is a consequence of his theory that genuine statements are pictures.
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  16. Essay on Russell on Modalities and Frege on Judgement.Shahid Rahman - forthcoming - History and Philosophy of Logic.
  17. Anselm and Russell.Maciej Nowicki - 2006 - Logic and Logical Philosophy 15 (4):355-368.
    In his paper “St. Anselm’s ontological argument succumbs to Russell’s paradox” Christopher Viger presents a critique of Anselm’s Argument from the second chapter of Proslogion. Viger claims there that he manages to show that the greater than relation that Anselm used in his proof leads to inconsistency. I argue firstly, that Viger does not show what he maintains to show, secondly, that the flaw is not in the nature of Anselm’s reasoning but in Viger’s (mis)understanding of Anselm as well as (...)
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  18. Review: Hugh J. Tallon, Russell's Doctrine of the Logical Proposition. [REVIEW]A. Wedberg - 1940 - Journal of Symbolic Logic 5 (2):74-74.
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  19. I Think, Therefore I Exist; I Belong, Therefore I Am.Juan José Luetich - 2012 - Transactions of The Luventicus Academy (3):1-4.
    The actions of perceiving and grouping are the two that the human being carries out when thinking in entities different from himself. In this article “The Mirror Problem” and “The Peer Problem”, which correspond respectively to self-perception and the perception of others, are studied. By solving these two problems, the thinker arrives to the following conclusions: “I exist” and “I am”.
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  20. Early Russell on Types and Plurals.Kevin C. Klement - 2014 - Journal for the History of Analytical Philosophy 2 (6):1-21.
    In 1903, in The Principles of Mathematics (PoM), Russell endorsed an account of classes whereupon a class fundamentally is to be considered many things, and not one, and used this thesis to explicate his first version of a theory of types, adding that it formed the logical justification for the grammatical distinction between singular and plural. The view, however, was short-lived; rejected before PoM even appeared in print. However, aside from mentions of a few misgivings, there is little evidence about (...)
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  21. The Foundations of Mathematics in the Theory of Sets.J. P. Mayberry - 2000 - Cambridge University Press.
    This book will appeal to mathematicians and philosophers interested in the foundations of mathematics.
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  22. The Consistency of Predicative Fragments of Frege's Grundgesetze der Arithmetik.Richard Heck - 1996 - History and Philosophy of Logic 17 (1):209-220.
    As is well-known, the formal system in which Frege works in his Grundgesetze der Arithmetik is formally inconsistent, Russell?s Paradox being derivable in it.This system is, except for minor differences, full second-order logic, augmented by a single non-logical axiom, Frege?s Axiom V. It has been known for some time now that the first-order fragment of the theory is consistent. The present paper establishes that both the simple and the ramified predicative second-order fragments are consistent, and that Robinson arithmetic, Q, is (...)
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  23. Idealist and Realist Elements in Cantor's Approach to Set Theory.I. Jane - 2010 - Philosophia Mathematica 18 (2):193-226.
    There is an apparent tension between the open-ended aspect of the ordinal sequence and the assumption that the set-theoretical universe is fully determinate. This tension is already present in Cantor, who stressed the incompletable character of the transfinite number sequence in Grundlagen and avowed the definiteness of the totality of sets and numbers in subsequent philosophical publications and in correspondence. The tension is particularly discernible in his late distinction between sets and inconsistent multiplicities. I discuss Cantor’s contrasting views, and I (...)
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  24. An Introduction to the Foundations and Fundamental Concepts of Mathematics.Howard Whitley Eves - 1958 - New York: Holt, Rinehart and Winston.
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  25. Bertrand Russell on His Paradox and the Multiplicative Axiom. An Unpublished Letter to Philip Jourdain.Ivor Grattan-Guinness - 1972 - Journal of Philosophical Logic 1 (2):103 - 110.
  26. Pluralism and the Foundations of Mathematics.Geoffrey Hellman - 2006 - In ¸ Itekellersetal:Sp. pp. 65--79.
    A plurality of approaches to foundational aspects of mathematics is a fact of life. Two loci of this are discussed here, the classicism/constructivism controversy over standards of proof, and the plurality of universes of discourse for mathematics arising in set theory and in category theory, whose problematic relationship is discussed. The first case illustrates the hypothesis that a sufficiently rich subject matter may require a multiplicity of approaches. The second case, while in some respects special to mathematics, raises issues of (...)
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  27. What is Categorical Structuralism?Geoffrey Hellman - 2006 - In Johan van Benthem, Gerhard Heinzman, M. Rebushi & H. Visser (eds.), The Age of Alternative Logics. Springer. pp. 151--161.
  28. Reflections on Skolem's Relativity of Set-Theoretical Concepts.Ignagio Jane - 2001 - Philosophia Mathematica 9 (2):129-153.
    In this paper an attempt is made to present Skolem's argument, for the relativity of some set-theoretical notions as a sensible one. Skolem's critique of set theory is seen as part of a larger argument to the effect that no conclusive evidence has been given for the existence of uncountable sets. Some replies to Skolem are discussed and are shown not to affect Skolem's position, since they all presuppose the existence of uncountable sets. The paper ends with an assessment of (...)
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  29. Philosophy, Mathematics, Science and Computation.Enrique V. Kortright - 1994 - Topoi 13 (1):51-60.
    Attempts to lay a foundation for the sciences based on modern mathematics are questioned. In particular, it is not clear that computer science should be based on set-theoretic mathematics. Set-theoretic mathematics has difficulties with its own foundations, making it reasonable to explore alternative foundations for the sciences. The role of computation within an alternative framework may prove to be of great potential in establishing a direction for the new field of computer science.Whitehead''s theory of reality is re-examined as a foundation (...)
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  30. The Consistency Problem for Set Theory: An Essay on the Cantorian Foundations of Mathematics (II).John Mayberry - 1977 - British Journal for the Philosophy of Science 28 (2):137-170.
  31. What is Required of a Foundation for Mathematics?John Mayberry - 1994 - Philosophia Mathematica 2 (1):16-35.
    The business of mathematics is definition and proof, and its foundations comprise the principles which govern them. Modern mathematics is founded upon set theory. In particular, both the axiomatic method and mathematical logic belong, by their very natures, to the theory of sets. Accordingly, foundational set theory is not, and cannot logically be, an axiomatic theory. Failure to grasp this point leads obly to confusion. The idea of a set is that of an extensional plurality, limited and definite in size, (...)
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  32. Sets, Wholes, and Limited Pluralitiest.Stephen Pollard - 1996 - Philosophia Mathematica 4 (1):42-58.
    This essay defends the following two claims: (1) liraitation-of-size reasoning yields enough sets to meet the needs of most mathematicians; (2) set formation and mereological fusion share enough logical features to justify placing both in the genus composition (even when the components of a set are taken to be its members rather than its subsets).
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  33. Foundations: Essays in Philosophy, Logic, Mathematics and Economics.Frank Plumpton Ramsey & D. H. Mellor (eds.) - 1978 - Humanties Press; Routledge.
  34. Review of G. Boolos, Logic, Logic, and Logic.Michael D. Resnik - 1999 - Philosophia Mathematica 7 (3):328-335.
Russell's Paradox
  1. The Completeness: From Henkin's Proposition to Quantum Computer.Vasil Penchev - 2018 - Логико-Философские Штудии 16 (1-2):134-135.
    The paper addresses Leon Hen.kin's proposition as a " lighthouse", which can elucidate a vast territory of knowledge uniformly: logic, set theory, information theory, and quantum mechanics: Two strategies to infinity are equally relevant for it is as universal and t hus complete as open and thus incomplete. Henkin's, Godel's, Robert Jeroslow's, and Hartley Rogers' proposition are reformulated so that both completeness and incompleteness to be unified and thus reduced as a joint property of infinity and of all infinite sets. (...)
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  2. Numbers, Empiricism and the A Priori.Olga Ramirez Calle - forthcoming - Logos and Episteme: An International Journal of Epistemology.
    The present paper deals with the ontological status of numbers and considers Frege ́s proposal in Grundlagen upon the background of the Post-Kantian semantic turn in analytical philosophy. Through a more systematic study of his philosophical premises, it comes to unearth a first level paradox that would unset earlier still than it was exposed by Russell. It then studies an alternative path, that departin1g from Frege’s initial premises, drives to a conception of numbers as synthetic a priori in a more (...)
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  3. The Significance of Evidence-Based Reasoning for Mathematics, Mathematics Education, Philosophy and the Natural Sciences.Bhupinder Singh Anand - manuscript
    In this multi-disciplinary investigation we show how an evidence-based perspective of quantification---in terms of algorithmic verifiability and algorithmic computability---admits evidence-based definitions of well-definedness and effective computability, which yield two unarguably constructive interpretations of the first-order Peano Arithmetic PA---over the structure N of the natural numbers---that are complementary, not contradictory. The first yields the weak, standard, interpretation of PA over N, which is well-defined with respect to assignments of algorithmically verifiable Tarskian truth values to the formulas of PA under the interpretation. (...)
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  4. Paradigms and Self-Reference: What is the Point of Asserting Paradoxical Sentences?Jakub Mácha - 2020 - In Shyam Wuppuluri & Newton da Costa (eds.), Wittgensteinian (adj.): Looking at the World from the Viewpoint of Wittgenstein's Philosophy. Cham, Švýcarsko: pp. 123-134.
    A paradox, according to Wittgenstein, is something surprising that is taken out of its context. Thus, one way of dealing with paradoxical sentences is to imagine the missing context of use. Wittgenstein formulates what I call the paradigm paradox: ‘one sentence can never describe the paradigm in another, unless it ceases to be a paradigm.’ (PG, p.346) There are several instances of this paradox scattered throughout Wittgenstein’s writings. I argue that this paradox is structurally equivalent to Russell’s paradox. The above (...)
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  5. Paradoxical Hypodoxes.Alexandre Billon - 2019 - Synthese 83 (12):5205-5229.
    Most paradoxes of self-reference have a dual or ‘hypodox’. The Liar paradox (Lr = ‘Lr is false’) has the Truth-Teller (Tt = ‘Tt is true’). Russell’s paradox, which involves the set of sets that are not self-membered, has a dual involving the set of sets which are self-membered, etc. It is widely believed that these duals are not paradoxical or at least not as paradoxical as the paradoxes of which they are duals. In this paper, I argue that some paradox’s (...)
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  6. The Barber Paradox: On its Paradoxicality and its Relationship to Russell's Paradox.Jiri Raclavsky - 2014 - Prolegomena 13 (2):269-278.
    The Barber paradox is often introduced as a popular version of Russell’s paradox, though some experts have denied their similarity, evencalling the Barber paradox a pseudoparadox. In the first part of thepaper, I demonstrate mainly that in the standard (Quinean) defini-tion of a paradox the Barber paradox is a clear-cut example of a non-paradox. Despite some outward similarities, it differs radically fromRussell’s paradox. I also expose many other differences. In the secondpart of the paper, I examine a probable source of (...)
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  7. Curry’s Paradox and Ω -Inconsistency.Andrew Bacon - 2013 - Studia Logica 101 (1):1-9.
    In recent years there has been a revitalised interest in non-classical solutions to the semantic paradoxes. In this paper I show that a number of logics are susceptible to a strengthened version of Curry's paradox. This can be adapted to provide a proof theoretic analysis of the omega-inconsistency in Lukasiewicz's continuum valued logic, allowing us to better evaluate which logics are suitable for a naïve truth theory. On this basis I identify two natural subsystems of Lukasiewicz logic which individually, but (...)
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  8. Solutions in the Origins of Math.Paul Bali - manuscript
    i. a poetic solution of the Goldbach Conjecture; ii. several responses to the Epimenides Paradox; iii. the volitional solution to Russell's Paradox.
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  9. Quantification and Paradox.Edward Ferrier - 2018 - Dissertation, University of Massachusetts Amherst
    I argue that absolutism, the view that absolutely unrestricted quantification is possible, is to blame for both the paradoxes that arise in naive set theory and variants of these paradoxes that arise in plural logic and in semantics. The solution is restrictivism, the view that absolutely unrestricted quantification is not possible. -/- It is generally thought that absolutism is true and that restrictivism is not only false, but inexpressible. As a result, the paradoxes are blamed, not on illicit quantification, but (...)
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  10. Philosophy of Logic. Hilary Putnam.John Corcoran - 1973 - Philosophy of Science 40 (1):131-133.
    Putnam, Hilary FPhilosophy of logic. Harper Essays in Philosophy. Harper Torchbooks, No. TB 1544. Harper & Row, Publishers, New York-London, 1971. v+76 pp. The author of this book has made highly regarded contributions to mathematics, to philosophy of logic and to philosophy of science, and in this book he brings his ideas in these three areas to bear on the traditional philosophic problem of materialism versus (objective) idealism. The book assumes that contemporary science (mathematical and physical) is largely correct as (...)
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  11. There is No Standard Model of ZFC.Jaykov Foukzon - 2018 - Journal of Global Research in Mathematical Archives 5 (1):33-50.
    Main results are:(i) Let M_st be standard model of ZFC. Then ~Con(ZFC+∃M_st), (ii) let k be an inaccessible cardinal then ~Con(ZFC+∃k),[10],11].
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  12. A Not So Fine Modal Version of Generality Relativism.Gonçalo Santos - 2010 - Theoria : An International Journal for Theory, History and Fundations of Science 25 (2):149-161.
    The generality relativist has been accused of holding a self-defeating thesis. Kit Fine proposed a modal version of generality relativism that tries to resist this claim. We discuss his proposal and argue that one of its formulations is self-defeating.
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  13. II-The Last Dogma of Type Confusions.Ofra Magidor - 2009 - Proceedings of the Aristotelian Society 109 (1pt1):1-29.
  14. Modal Set Theory.Christopher Menzel - forthcoming - In Otávio Bueno & Scott Shalkowski (eds.), The Routledge Handbook of Modality. London and 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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  15. Maximally Consistent Sets of Instances of Naive Comprehension.Luca Incurvati & Julien Murzi - 2017 - Mind 126 (502).
    Paul Horwich (1990) once suggested restricting the T-Schema to the maximally consistent set of its instances. But Vann McGee (1992) proved that there are multiple incompatible such sets, none of which, given minimal assumptions, is recursively axiomatizable. The analogous view for set theory---that Naïve Comprehension should be restricted according to consistency maxims---has recently been defended by Laurence Goldstein (2006; 2013). It can be traced back to W.V.O. Quine(1951), who held that Naïve Comprehension embodies the only really intuitive conception of set (...)
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  16. Contradictions Inherent in Special Relativity: Space Varies.Kim Joosoak - manuscript
    Special relativity has changed the fundamental view on space and time since Einstein introduced it in 1905. It substitutes four dimensional spacetime for the absolute space and time of Newtonian mechanics. It is believed that the validities of Lorentz invariants are fully confirmed empirically for the last one hundred years and therefore its status are canonical underlying all physical principles. However, spacetime metric is a geometric approach on nature when we interpret the natural phenomenon. A geometric flaw on this will (...)
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