This category needs an editor. We encourage you to help if you are qualified.
Volunteer, or read more about what this involves.
Related categories

52 found
Order:
1 — 50 / 52
  1. Modality and Hyperintensionality in Mathematics.Hasen Khudairi - manuscript
    This paper aims to contribute to the analysis of the nature of mathematical modality, and to the applications of the latter to unrestricted quantification and absolute decidability. Rather than countenancing the interpretational type of mathematical modality as a primitive, I argue that the interpretational type of mathematical modality is a species of epistemic modality. I argue, then, that the framework of two-dimensional semantics ought to be applied to the mathematical setting. The framework permits of a formally precise account of the (...)
    Remove from this list   Direct download  
    Translate
     
     
    Export citation  
     
    Bookmark  
  2. Retrieving the Mathematical Mission of the Continuum Concept From the Transfinitely Reductionist Debris of Cantor’s Paradise. Extended Abstract.Edward G. Belaga - forthcoming - International Journal of Pure and Applied Mathematics.
    What is so special and mysterious about the Continuum, this ancient, always topical, and alongside the concept of integers, most intuitively transparent and omnipresent conceptual and formal medium for mathematical constructions and the battle field of mathematical inquiries ? And why it resists the century long siege by best mathematical minds of all times committed to penetrate once and for all its set-theoretical enigma ? -/- The double-edged purpose of the present study is to save from the transfinite deadlock of (...)
    Remove from this list   Direct download  
    Translate
     
     
    Export citation  
     
    Bookmark  
  3. In Defense of Countabilism.David Builes & Jessica M. Wilson - forthcoming - Philosophical Studies:1-38.
    Inspired by Cantor's Theorem (CT), orthodoxy takes infinities to come in different sizes. The orthodox view has had enormous influence in mathematics, philosophy, and science. We will defend the contrary view---Countablism---according to which, necessarily, every infinite collection (set or plurality) is countable. We first argue that the potentialist or modal strategy for treating Russell's Paradox, first proposed by Parsons (2000) and developed by Linnebo (2010, 2013) and Linnebo and Shapiro (2019), should also be applied to CT, in a way that (...)
    Remove from this list   Direct download (4 more)  
     
    Export citation  
     
    Bookmark  
  4. The Price of Mathematical Scepticism.Paul Blain Levy - forthcoming - Philosophia Mathematica.
    This paper argues that, insofar as we doubt the bivalence of the Continuum Hypothesis or the truth of the Axiom of Choice, we should also doubt the consistency of third-order arithmetic, both the classical and intuitionistic versions. -/- Underlying this argument is the following philosophical view. Mathematical belief springs from certain intuitions, each of which can be either accepted or doubted in its entirety, but not half-accepted. Therefore, our beliefs about reality, bivalence, choice and consistency should all be aligned.
    Remove from this list   Direct download  
     
    Export citation  
     
    Bookmark  
  5. Chance and the Continuum Hypothesis.Daniel Hoek - 2021 - 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 (...)
    Remove from this list   Direct download (5 more)  
     
    Export citation  
     
    Bookmark   3 citations  
  6. Maximality Principles in Set Theory.Luca Incurvati - 2017 - Philosophia Mathematica 25 (2):159-193.
    In set theory, a maximality principle is a principle that asserts some maximality property of the universe of sets or some part thereof. Set theorists have formulated a variety of maximality principles in order to settle statements left undecided by current standard set theory. In addition, philosophers of mathematics have explored maximality principles whilst attempting to prove categoricity theorems for set theory or providing criteria for selecting foundational theories. This article reviews recent work concerned with the formulation, investigation and justification (...)
    Remove from this list   Direct download (4 more)  
     
    Export citation  
     
    Bookmark   5 citations  
  7. Numerical Infinities and Infinitesimals: Methodology, Applications, and Repercussions on Two Hilbert Problems.Yaroslav Sergeyev - 2017 - EMS Surveys in Mathematical Sciences 4 (2):219–320.
    In this survey, a recent computational methodology paying a special attention to the separation of mathematical objects from numeral systems involved in their representation is described. It has been introduced with the intention to allow one to work with infinities and infinitesimals numerically in a unique computational framework in all the situations requiring these notions. The methodology does not contradict Cantor’s and non-standard analysis views and is based on the Euclid’s Common Notion no. 5 “The whole is greater than the (...)
    Remove from this list   Direct download (2 more)  
     
    Export citation  
     
    Bookmark   2 citations  
  8. 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.
    Remove from this list   Direct download (3 more)  
     
    Export citation  
     
    Bookmark   11 citations  
  9. Naive Infinitism: The Case for an Inconsistency Approach to Infinite Collections.Toby Meadows - 2015 - Notre Dame Journal of Formal Logic 56 (1):191-212.
    This paper expands upon a way in which we might rationally doubt that there are multiple sizes of infinity. The argument draws its inspiration from recent work in the philosophy of truth and philosophy of set theory. More specifically, elements of contextualist theories of truth and multiverse accounts of set theory are brought together in an effort to make sense of Cantor’s troubling theorem. The resultant theory provides an alternative philosophical perspective on the transfinite, but has limited impact on everyday (...)
    Remove from this list   Direct download (7 more)  
     
    Export citation  
     
    Bookmark   5 citations  
  10. UN SEMPLICE MODO PER TRATTARE LE GRANDEZZE INFINITE ED INFINITESIME.Yaroslav Sergeyev - 2015 - la Matematica Nella Società E Nella Cultura: Rivista Dell’Unione Matematica Italiana, Serie I 8:111-147.
    A new computational methodology allowing one to work in a new way with infinities and infinitesimals is presented in this paper. The new approach, among other things, gives the possibility to calculate the number of elements of certain infinite sets, avoids indeterminate forms and various kinds of divergences. This methodology has been used by the author as a starting point in developing a new kind of computer – the Infinity Computer – able to execute computations and to store in its (...)
    Remove from this list   Direct download  
    Translate
     
     
    Export citation  
     
    Bookmark   3 citations  
  11. What is Absolute Undecidability?†.Justin Clarke-Doane - 2013 - Noûs 47 (3):467-481.
    It is often alleged that, unlike typical axioms of mathematics, the Continuum Hypothesis (CH) is indeterminate. This position is normally defended on the ground that the CH is undecidable in a way that typical axioms are not. Call this kind of undecidability “absolute undecidability”. In this paper, I seek to understand what absolute undecidability could be such that one might hope to establish that (a) CH is absolutely undecidable, (b) typical axioms are not absolutely undecidable, and (c) if a mathematical (...)
    Remove from this list   Direct download (3 more)  
     
    Export citation  
     
    Bookmark   11 citations  
  12. The Set-Theoretic Multiverse.Joel David Hamkins - 2012 - Review of Symbolic Logic 5 (3):416-449.
    The multiverse view in set theory, introduced and argued for in this article, is the view that there are many distinct concepts of set, each instantiated in a corresponding set-theoretic universe. The universe view, in contrast, asserts that there is an absolute background set concept, with a corresponding absolute set-theoretic universe in which every set-theoretic question has a definite answer. The multiverse position, I argue, explains our experience with the enormous range of set-theoretic possibilities, a phenomenon that challenges the universe (...)
    Remove from this list   Direct download (6 more)  
     
    Export citation  
     
    Bookmark   73 citations  
  13. Set Theory: Boolean-Valued Models and Independence Proofs.John L. Bell - 2011 - Oxford University Press.
    This third edition, now available in paperback, is a follow up to the author's classic Boolean-Valued Models and Independence Proofs in Set Theory. It provides an exposition of some of the most important results in set theory obtained in the 20th century: the independence of the continuum hypothesis and the axiom of choice.
    Remove from this list   Direct download  
     
    Export citation  
     
    Bookmark   13 citations  
  14. A Natural Model of the Multiverse Axioms.Victoria Gitman & Joel David Hamkins - 2010 - Notre Dame Journal of Formal Logic 51 (4):475-484.
    If ZFC is consistent, then the collection of countable computably saturated models of ZFC satisfies all of the Multiverse Axioms of Hamkins.
    Remove from this list   Direct download (9 more)  
     
    Export citation  
     
    Bookmark   10 citations  
  15. On the Question of Absolute Undecidability.Peter Koellner - 2010 - In Kurt Gödel, Solomon Feferman, Charles Parsons & Stephen G. Simpson (eds.), Philosophia Mathematica. Association for Symbolic Logic. pp. 153-188.
    The paper begins with an examination of Gödel's views on absolute undecidability and related topics in set theory. These views are sharpened and assessed in light of recent developments. It is argued that a convincing case can be made for axioms that settle many of the questions undecided by the standard axioms and that in a precise sense the program for large cardinals is a complete success “below” CH. It is also argued that there are reasonable scenarios for settling CH (...)
    Remove from this list   Direct download (7 more)  
     
    Export citation  
     
    Bookmark   29 citations  
  16. The Search for Diamonds: Review of S. Shelah, Middle Diamond; S. Shelah, Diamonds; and M. Zeman, Diamond, GCH and Weak Square. [REVIEW]Assaf Rinot - 2010 - Bulletin of Symbolic Logic 16 (3):420 - 423.
  17. On the Reality of the Continuum Discussion Note: A Reply to Ormell, ‘Russell's Moment of Candour’, Philosophy: Anne Newstead and James Franklin.Anne Newstead - 2008 - Philosophy 83 (1):117-127.
    In a recent article, Christopher Ormell argues against the traditional mathematical view that the real numbers form an uncountably infinite set. He rejects the conclusion of Cantor’s diagonal argument for the higher, non-denumerable infinity of the real numbers. He does so on the basis that the classical conception of a real number is mys- terious, ineffable, and epistemically suspect. Instead, he urges that mathematics should admit only ‘well-defined’ real numbers as proper objects of study. In practice, this means excluding as (...)
    Remove from this list   Direct download (4 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  18. Individui e continui.Venanzio Raspa - 2008 - Rivista di Estetica 39:189-214.
    Starting with the philosophical reflections of the Italian writer C. E. Gadda, the paper offers a criticism of the traditional concept of an individual as something which is determinate, separate and autonomous. Gadda argues that an individual should be understood as an element which is in a multiplicity of relations with the other elements of the system inside of which it exists. The idea is developed on the basis of Spinoza's 'Ethics', but it shares many affinities with Peirce's notions of (...)
    Remove from this list   Direct download (3 more)  
    Translate
     
     
    Export citation  
     
    Bookmark   5 citations  
  19. Synechism: The Keystone of Peirce's Metaphysics.Joseph Esposito - 2005 - The Commens Encyclopedia: The Digital Encyclopedia of Peirce Studies.
    Synechism, as a metaphysical theory, is the view that the universe exists as a continuous whole of all of its parts, with no part being fully separate, determined or determinate, and continues to increase in complexity and connectedness through semiosis and the operation of an irreducible and ubiquitous power of relational generality to mediate and unify substrates. As a research program, synechism is a scientific maxim to seek continuities where discontinuities are thought to be permanent and to seek semiotic relations (...)
    Remove from this list   Direct download  
     
    Export citation  
     
    Bookmark   5 citations  
  20. Absorption, Hallucinations, and the Continuum Hypothesis.Joseph Glicksohn - 2004 - Behavioral and Brain Sciences 27 (6):793-794.
    The target article, in stressing the balance between neurobiological and psychological factors, makes a compelling argument in support of a continuum of perceptual and hallucinatory experience. Nevertheless, two points need to be addressed. First, the authors are probably underestimating the incidence of hallucinations in the normal population. Second, one should consider the role of absorption as a predisposing factor for hallucinations.
    Remove from this list   Direct download (8 more)  
     
    Export citation  
     
    Bookmark   3 citations  
  21. Review of J. Cummings, A Model in Which GCH Holds at Successors but Fails at Limits; Strong Ultrapowers and Long Core Models; Coherent Sequences Versus Radin Sequences; and J. Cummings, M. Foreman, and M. Magidor, Squares, Scales and Stationary Reflection. [REVIEW]Arthur W. Apter - 2002 - Bulletin of Symbolic Logic 8 (4):550-552.
    Remove from this list   Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  22. Is Cantor's Continuum Problem Inherently Vague?Kai Hauser - 2002 - Philosophia Mathematica 10 (3):257-285.
    I examine various claims to the effect that Cantor's Continuum Hypothesis and other problems of higher set theory are ill-posed questions. The analysis takes into account the viability of the underlying philosophical views and recent mathematical developments.
    Remove from this list   Direct download (9 more)  
     
    Export citation  
     
    Bookmark   11 citations  
  23. Multiple Universes of Sets and Indeterminate Truth Values.Donald A. Martin - 2001 - Topoi 20 (1):5-16.
  24. Preservation Theorems Without Continuum Hypothesis.George C. Nelson - 1998 - Studia Logica 60 (3):343-355.
    Many results concerning the equivalence between a syntactic form of formulas and a model theoretic conditions are proven directly without using any form of a continuum hypothesis. In particular, it is demonstrated that any reduced product sentence is equivalent to a Horn sentence. Moreover, in any first order language without equality one now has that a reduced product sentence is equivalent to a Horn sentence and any sentence is equivalent to a Boolean combination of Horn sentences.
    Remove from this list   Direct download (5 more)  
     
    Export citation  
     
    Bookmark  
  25. Some Logical Remarks Concerning the Continuum Problem.C. Alvarez Jimenez - 1995 - Boston Studies in the Philosophy of Science 172:173-186.
    Remove from this list  
     
    Export citation  
     
    Bookmark  
  26. Axioms of Symmetry: Throwing Darts at the Real Number Line.Chris Freiling - 1986 - Journal of Symbolic Logic 51 (1):190-200.
    We will give a simple philosophical "proof" of the negation of Cantor's continuum hypothesis (CH). (A formal proof for or against CH from the axioms of ZFC is impossible; see Cohen [1].) We will assume the axioms of ZFC together with intuitively clear axioms which are based on some intuition of Stuart Davidson and an old theorem of Sierpinski and are justified by the symmetry in a thought experiment throwing darts at the real number line. We will in fact show (...)
    Remove from this list   Direct download (8 more)  
     
    Export citation  
     
    Bookmark   25 citations  
  27. On Forcing Without the Continuum Hypothesis.Uri Abraham - 1983 - Journal of Symbolic Logic 48 (3):658-661.
    Remove from this list   Direct download (7 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  28. The Continuum Hypothesis in Intuitionism.W. Gielen, H. de Swart & W. Veldman - 1981 - Journal of Symbolic Logic 46 (1):121-136.
    Remove from this list   Direct download (8 more)  
     
    Export citation  
     
    Bookmark   5 citations  
  29. Georg Cantor: The Personal Matrix of His Mathematics.Joseph Dauben - 1978 - Isis 69:534-550.
  30. A Note on the Axiom of Choice and the Continuum Hypothesis.Rolf Schock - 1977 - Notre Dame Journal of Formal Logic 18 (3):409-414.
  31. The Continuum Hypothesis is Independent of Second-Order ZF.Thomas S. Weston - 1977 - Notre Dame Journal of Formal Logic 18 (3):499-503.
    Remove from this list   Direct download (5 more)  
     
    Export citation  
     
    Bookmark   2 citations  
  32. Higher Souslin Trees and the Generalized Continuum Hypothesis.John Gregory - 1976 - Journal of Symbolic Logic 41 (3):663-671.
  33. Kreisel, the Continuum Hypothesis and Second Order Set Theory.Thomas Weston - 1976 - Journal of Philosophical Logic 5 (2):281 - 298.
    The major point of contention among the philosophers and mathematicians who have written about the independence results for the continuum hypothesis (CH) and related questions in set theory has been the question of whether these results give reason to doubt that the independent statements have definite truth values. This paper concerns the views of G. Kreisel, who gives arguments based on second order logic that the CH does have a truth value. The view defended here is that although Kreisel's conclusion (...)
    Remove from this list   Direct download (4 more)  
     
    Export citation  
     
    Bookmark   19 citations  
  34. The Generalized Continuum Hypothesis is Equivalent to the Generalized Maximization Principle.Joel I. Friedman - 1971 - Journal of Symbolic Logic 36 (1):39-54.
    In spite of the work of Gödel and Cohen, which showed the undecidability of the Generalized Continuum Hypothesis from the axioms of set theory, the problem still remains to decide GCH on the basis of new axioms. It is almost 100 years since Cantor first conjectured the Continuum Hypothesis, yet we seem to be no closer to determining its truth. Nevertheless, it is a sound methodological principle that given any undecidable set-theoretical statement, we should search for “other axioms of set (...)
    Remove from this list   Direct download (8 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  35. Eliminating the Continuum Hypothesis.Richard A. Platek - 1969 - Journal of Symbolic Logic 34 (2):219-225.
    Remove from this list   Direct download (7 more)  
     
    Export citation  
     
    Bookmark  
  36. Set Theory and the Continuum Hypothesis.Paul J. Cohen - 1966 - New York: W. A. Benjamin.
    This exploration of a notorious mathematical problem is the work of the man who discovered the solution. Written by an award-winning professor at Stanford University, it employs intuitive explanations as well as detailed mathematical proofs in a self-contained treatment. This unique text and reference is suitable for students and professionals. 1966 edition. Copyright renewed 1994.
    Remove from this list   Direct download (2 more)  
     
    Export citation  
     
    Bookmark   100 citations  
  37. A Simple Version of the Generalized Continuum Hypothesis.Rolf Schock - 1966 - Notre Dame Journal of Formal Logic 7 (3):287-288.
    Remove from this list   Direct download (4 more)  
     
    Export citation  
     
    Bookmark  
  38. The Independence of the Continuum Hypothesis.Paul J. Cohen - 1963 - Proceedings of the National Academy of Sciences of the United States of America 50 (6):1143--8.
    Remove from this list   Direct download  
     
    Export citation  
     
    Bookmark   44 citations  
  39. On Gödel's Proof That $V=L$ Implies the Generalized Continuum Hypothesis.Raouf Doss - 1963 - Notre Dame Journal of Formal Logic 4 (4):283-287.
    Remove from this list   Direct download (4 more)  
     
    Export citation  
     
    Bookmark  
  40. A Note on the Generalized Continuum Hypothesis. III.Bolesław Sobociński - 1963 - Notre Dame Journal of Formal Logic 4 (3):233-240.
    Remove from this list   Direct download (3 more)  
     
    Export citation  
     
    Bookmark  
  41. A Note on the Generalized Continuum Hypothesis. II.Bolesław Sobociński - 1963 - Notre Dame Journal of Formal Logic 4 (1):67-79.
    Remove from this list   Direct download (5 more)  
     
    Export citation  
     
    Bookmark  
  42. A Note On The Generalized Continuum Hypothesis, Ii.Bolesław Sobociński - 1963 - Notre Dame Journal of Formal Logic 4 (1):67-79.
    Remove from this list   Direct download (5 more)  
     
    Export citation  
     
    Bookmark  
  43. A Note on the Generalized Continuum Hypothesis. I.Bolesław Sobociński - 1962 - Notre Dame Journal of Formal Logic 3 (4):274-278.
    Remove from this list   Direct download (4 more)  
     
    Export citation  
     
    Bookmark  
  44. Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis with the Axioms of Set Theory.Kurt Gödel - 1940 - Princeton, NJ, USA: Princeton University Press.
  45. Consistency of the Continuum Hypothesis.Kurt Gödel - 1940 - Princeton University Press;.
    Kurt Gödel, mathematician and logician, was one of the most influential thinkers of the twentieth century. Gödel fled Nazi Germany, fearing for his Jewish wife and fed up with Nazi interference in the affairs of the mathematics institute at the University of Göttingen. In 1933 he settled at the Institute for Advanced Study in Princeton, where he joined the group of world-famous mathematicians who made up its original faculty. His 1940 book, better known by its short title, The Consistency of (...)
    Remove from this list   Direct download (2 more)  
     
    Export citation  
     
    Bookmark   62 citations  
  46. The Consistency of the Continuum Hypothesis.Kurt Godel - 1940 - Princeton University Press.
    Previously published: Princeton University Press, 1940.
    Remove from this list   Direct download  
     
    Export citation  
     
    Bookmark   39 citations  
  47. Indeterminacy.Prentice Hall - unknown
    It is well known that, for example, the Continuum Hypothesis can’t be proved or disproved from the standard axioms of set theory or their familiar extensions. Some think it follows that CH has no determinate truth value; others insist that this conclusion is false, not because there is some objective world of sets in which CH is either true or false, but on logical grounds. Claims of indeterminacy have also been made on the basis of such considerations as the existence (...)
    Remove from this list   Direct download  
     
    Export citation  
     
    Bookmark  
  48. Is the Continuum Hypothesis True, False, or Neither?David J. Chalmers - manuscript
    Thanks to all the people who responded to my enquiry about the status of the Continuum Hypothesis. This is a really fascinating subject, which I could waste far too much time on. The following is a summary of some aspects of the feeling I got for the problems. This will be old hat to set theorists, and no doubt there are a couple of embarrassing misunderstandings, but it might be of some interest to non professionals.
    Remove from this list   Direct download  
     
    Export citation  
     
    Bookmark  
  49. Conceptions of the Continuum.Solomon Feferman - unknown
    Key words: the continuum, structuralism, conceptual structuralism, basic structural conceptions, Euclidean geometry, Hilbertian geometry, the real number system, settheoretical conceptions, phenomenological conceptions, foundational conceptions, physical conceptions.
    Remove from this list   Direct download (2 more)  
     
    Export citation  
     
    Bookmark   20 citations  
  50. Conceptual Structuralism and the Continuum.Solomon Feferman - unknown
    • This comes from my general view of the nature of mathematics, that it is humanly based and that it deals with more or less clear conceptions of mathematical structures; for want of a better word, I call that view conceptual structuralism.
    Remove from this list   Direct download  
     
    Export citation  
     
    Bookmark   1 citation  
1 — 50 / 52