25 found
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  1. Cantor’s Absolute in Metaphysics and Mathematics.Kai Hauser - 2013 - International Philosophical Quarterly 53 (2):161-188.
    This paper explores the metaphysical roots of Cantor’s conception of absolute infinity in order to shed some light on two basic issues that also affect the mathematical theory of sets: the viability of Cantor’s distinction between sets and inconsistent multiplicities, and the intrinsic justification of strong axioms of infinity that are studied in contemporary set theory.
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  2. Gödel's Program Revisited Part I: The Turn to Phenomenology.Kai Hauser - 2006 - Bulletin of Symbolic Logic 12 (4):529-590.
    Convinced that the classically undecidable problems of mathematics possess determinate truth values, Gödel issued a programmatic call to search for new axioms for their solution. The platonism underlying his belief in the determinateness of those questions in combination with his conception of intuition as a kind of perception have struck many of his readers as highly problematic. Following Gödel's own suggestion, this article explores ideas from phenomenology to specify a meaning for his mathematical realism that allows for a defensible epistemology.
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  3. 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.
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  4. Lotze and Husserl.Kai Hauser - 2003 - Archiv für Geschichte der Philosophie 85 (2):152-178.
  5.  64
    Indescribable Cardinals and Elementary Embeddings.Kai Hauser - 1991 - Journal of Symbolic Logic 56 (2):439-457.
  6.  28
    The Consistency Strength of Projective Absoluteness.Kai Hauser - 1995 - Annals of Pure and Applied Logic 74 (3):245-295.
    It is proved that in the absence of proper class inner models with Woodin cardinals, for each n ε {1,…,ω}, ∑3 + n1 absoluteness implies there are n strong cardinals in K (where this denotes a suitably defined global version of the core model for one Woodin cardinal as exposed by Steel. Combined with a forcing argument of Woodin, this establishes that the consistency strength of ∑3 + n1 absoluteness is exactly that of n strong cardinals so that in particular (...)
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  7. Strong Axioms of Infinity and the Debate About Realism.Kai Hauser & W. Hugh Woodin - 2014 - Journal of Philosophy 111 (8):397-419.
    One of the most distinctive and intriguing developments of modern set theory has been the realization that, despite widely divergent incentives for strengthening the standard axioms, there is essentially only one way of ascending the higher reaches of infinity. To the mathematical realist the unexpected convergence suggests that all these axiomatic extensions describe different aspects of the same underlying reality.
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  8.  28
    Perception, Intuition, and Reliability.Kai Hauser & Tahsİn Öner - 2018 - Theoria 84 (1):23-59.
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  9.  46
    Is Choice Self-Evident?Kai Hauser - 2005 - American Philosophical Quarterly 42 (4):237 - 261.
  10.  16
    Projective Uniformization Revisited.Kai Hauser & Ralf-Dieter Schindler - 2000 - Annals of Pure and Applied Logic 103 (1-3):109-153.
    We give an optimal lower bound in terms of large cardinal axioms for the logical strength of projective uniformization in conjuction with other regularity properties of projective sets of real numbers, namely Lebesgue measurability and its dual in the sense of category . Our proof uses a projective computation of the real numbers which code inital segments of a core model and answers a question in Hauser.
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  11.  49
    Indescribable Cardinals Without Diamonds.Kai Hauser - 1992 - Archive for Mathematical Logic 31 (5):373-383.
    We show that form, n≧1 the existence of a∏ n m indescribable cardinal is equiconsistent with the failure of the combinatorial principle at a∏ n m indescribable cardinal κ together with the Generalized Continuum Hypothesis.
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  12.  11
    Strong Cardinals in the Core Model.Kai Hauser & Greg Hjorth - 1997 - Annals of Pure and Applied Logic 83 (2):165-198.
  13.  57
    Objectivity Over Objects: A Case Study in Theory Formation.Kai Hauser - 2001 - Synthese 128 (3):245 - 285.
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  14.  77
    What New Axioms Could Not Be.Kai Hauser - 2002 - Dialectica 56 (2):109–124.
    The paper exposes the philosophical and mathematical flaws in an attempt to settle the continuum problem by a new class of axioms based on probabilistic reasoning. I also examine the larger proposal behind this approach, namely the introduction of new primitive notions that would supersede the set theoretic foundation of mathematics.
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  15. Cantor’s Concept of Set in the Light of Plato’s Philebus.Kai Hauser - 2010 - Review of Metaphysics 63 (4):783-805.
    In explaining his concept of set Cantor intimates a connection with the metaphysical scheme put forward in Plato’s Philebus to determine the place of pleasure. We argue that these determinations capture key ideas of Cantorian set theory and, moreover, extend to intuitions which continue to play a central role in the modern mathematics of infinity.
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  16.  21
    Generic Relativizations of Fine Structure.Kai Hauser - 2000 - Archive for Mathematical Logic 39 (4):227-251.
    It is shown how certain generic extensions of a fine structural model in the sense of Mitchell and Steel [MiSt] can be reorganized as relativizations of the model to the generic object. This is then applied to the construction of Steel's core model for one Woodin cardinal [St] and its generalizations.
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  17.  29
    A Minimal Counterexample to Universal Baireness.Kai Hauser - 1999 - Journal of Symbolic Logic 64 (4):1601-1627.
    For a canonical model of set theory whose projective theory of the real numbers is stable under set forcing extensions, a set of reals of minimal complexity is constructed which fails to be universally Baire. The construction uses a general method for generating non-universally Baire sets via the Levy collapse of a cardinal, as well as core model techniques. Along the way it is shown (extending previous results of Steel) how sufficiently iterable fine structure models recognize themselves as global core (...)
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  18.  8
    Objectivity Over Objects: A Case Study In Theory Formation.Kai Hauser - 2001 - Synthese 128 (3):245-285.
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  19.  5
    A Minimal Counterexample To Universal Baireness.Kai Hauser - 1999 - Journal of Symbolic Logic 64 (4):1601-1627.
    For a canonical model of set theory whose projective theory of the real numbers is stable under set forcing extensions, a set of reals of minimal complexity is constructed which fails to be universally Baire. The construction uses a general method for generating non-universally Baire sets via the Levy collapse of a cardinal, as well as core model techniques. Along the way it is shown how sufficiently iterable fine structure models recognize themselves as global core models.
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  20.  6
    The Indescribability of the Order of the Indescribable Cardinals.Kai Hauser - 1992 - Annals of Pure and Applied Logic 57 (1):45-91.
    We prove the following consistency results about indescribable cardinals which answer a question of A. Kanamori and M. Magidor .Theorem 1.1 . CON.Theorem 5.1 . Assuming the existence of σmn indescribable cardinals for all m < ω and n < ω and given a function : {: m 2, n } 1} → {0,1} there is a poset P L[] such that GCH holds in P and Theorem 1.1 extends the work begun in [2], and its proof uses an iterated (...)
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  21.  18
    Erratum To: Intuition and Its Object.Kai Hauser - 2015 - Axiomathes 25 (3):283-284.
    Erratum to: Axiomathes DOI 10.1007/s10516-014-9234-yIn the original publication of the article, some of the references were published incorrectly. Please find below the corrected version of these references.
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  22.  99
    Intuition and Its Object.Kai Hauser - 2015 - Axiomathes 25 (3):253-281.
    The view that mathematics deals with ideal objects to which we have epistemic access by a kind of perception has troubled many thinkers. Using ideas from Husserl’s phenomenology, I will take a different look at these matters. The upshot of this approach is that there are non-material objects and that they can be recognized in a process very closely related to sense perception. In fact, the perception of physical objects may be regarded as a special case of this more universal (...)
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  23.  43
    Sets and Singletons.Kai Hauser & W. Hugh Woodin - 1999 - Journal of Symbolic Logic 64 (2):590-616.
    We extend work of H. Friedman, L. Harrington and P. Welch to the third level of the projective hierarchy. Our main theorems say that (under appropriate background assumptions) the possibility to select definable elements of non-empty sets of reals at the third level of the projective hierarchy is equivalent to the disjunction of determinacy of games at the second level of the projective hierarchy and the existence of a core model (corresponding to this fragment of determinacy) which must then contain (...)
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  24.  5
    What New Axioms Could Not Be.Kai Hauser - 2002 - Dialectica 56 (2):109-124.
    The paper exposes the philosophical and mathematical flaws in an attempt to settle the continuum problem by a new class of axioms based on probabilistic reasoning. I also examine the larger proposal behind this approach, namely the introduction of new primitive notions that would supersede the set theoretic foundation of mathematics.
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  25. REVIEWS-Phenomenology, Logic, and the Philosophy of Mathematics.R. Tieszen & Kai Hauser - 2007 - Bulletin of Symbolic Logic 13 (3):365-367.
    Offering a collection of fifteen essays that deal with issues at the intersection of phenomenology, logic, and the philosophy of mathematics, this 2005 book is divided into three parts. Part I contains a general essay on Husserl's conception of science and logic, an essay of mathematics and transcendental phenomenology, and an essay on phenomenology and modern pure geometry. Part II is focused on Kurt Godel's interest in phenomenology. It explores Godel's ideas and also some work of Quine, Penelope Maddy and (...)
     
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