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W. Hugh Woodin [26]W. H. Woodin [11]
  1.  51
    In Search of Ultimate-L the 19th Midrasha Mathematicae Lectures.W. Hugh Woodin - 2017 - Bulletin of Symbolic Logic 23 (1):1-109.
    We give a fairly complete account which first shows that the solution to the inner model problem for one supercompact cardinal will yield an ultimate version ofLand then shows that the various current approaches to inner model theory must be fundamentally altered to provide that solution.
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  2.  20
    The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal.W. Hugh Woodin - 2002 - Bulletin of Symbolic Logic 8 (1):91-93.
  3.  55
    Suitable Extender Models I.W. Hugh Woodin - 2010 - Journal of Mathematical Logic 10 (1):101-339.
    We investigate both iteration hypotheses and extender models at the level of one supercompact cardinal. The HOD Conjecture is introduced and shown to be a key conjecture both for the Inner Model Program and for understanding the limits of the large cardinal hierarchy. We show that if the HOD Conjecture is true then this provides strong evidence for the existence of an ultimate version of Gödel's constructible universe L. Whether or not this "ultimate" L exists is now arguably the central (...)
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  4.  87
    Suitable Extender Models II: Beyond Ω-Huge.W. Hugh Woodin - 2011 - Journal of Mathematical Logic 11 (2):115-436.
    We investigate large cardinal axioms beyond the level of ω-huge in context of the universality of the suitable extender models of [Suitable Extender Models I, J. Math. Log.10 101–339]. We show that there is an analog of ADℝ at the level of ω-huge, more precisely the construction of the minimum model of ADℝ generalizes to the level of Vλ+1. This allows us to formulate the indicated generalization of ADℝ and then to prove that if the axiom holds in V at (...)
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  5.  26
    Large Cardinals Beyond Choice.Joan Bagaria, Peter Koellner & W. Hugh Woodin - 2019 - Bulletin of Symbolic Logic 25 (3):283-318.
    The HOD Dichotomy Theorem states that if there is an extendible cardinal, δ, then either HOD is “close” to V or HOD is “far” from V. The question is whether the future will lead to the first or the second side of the dichotomy. Is HOD “close” to V, or “far” from V? There is a program aimed at establishing the first alternative—the “close” side of the HOD Dichotomy. This is the program of inner model theory. In recent years the (...)
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  6. $\Underset{\Tilde}{\Delta}^1_n$ Sets Of Reals.Joan Bagaria & W. Hugh Woodin - 1997 - Journal of Symbolic Logic 62 (4):1379-1428.
  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.  74
    The Necessary Maximality Principle for C. C. C. Forcing is Equiconsistent with a Weakly Compact Cardinal.Joel D. Hamkins & W. Hugh Woodin - 2005 - Mathematical Logic Quarterly 51 (5):493-498.
    The Necessary Maximality Principle for c. c. c. forcing with real parameters is equiconsistent with the existence of a weakly compact cardinal.
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  9.  64
    On the Consistency Strength of the Inner Model Hypothesis.Sy-David Friedman, Philip Welch & W. Hugh Woodin - 2008 - Journal of Symbolic Logic 73 (2):391 - 400.
  10.  36
    The Transfinite Universe.W. Hugh Woodin - 2011 - In Matthias Baaz (ed.), Kurt Gödel and the Foundations of Mathematics: Horizons of Truth. Cambridge University Press. pp. 449.
  11.  27
    Definability in the Enumeration Degrees.Theodore A. Slaman & W. Hugh Woodin - 1997 - Archive for Mathematical Logic 36 (4-5):255-267.
    . We prove that every countable relation on the enumeration degrees, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} ${\frak E}$\end{document}, is uniformly definable from parameters in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} ${\frak E}$\end{document}. Consequently, the first order theory of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} ${\frak E}$\end{document} is recursively isomorphic to the second order theory of arithmetic. By an effective version of coding lemma, we show that the first (...)
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  12.  61
    Incompatible Ω-Complete Theories.Peter Koellner & W. Hugh Woodin - 2009 - Journal of Symbolic Logic 74 (4):1155 - 1170.
    In 1985 the second author showed that if there is a proper class of measurable Woodin cardinals and $V^{B1} $ and $V^{B2} $ are generic extensions of V satisfying CH then $V^{B1} $ and $V^{B2} $ agree on all $\Sigma _1^2 $ -statements. In terms of the strong logic Ω-logic this can be reformulated by saying that under the above large cardinal assumption ZFC + CH is Ω-complete for $\Sigma _1^2 $ Moreover. CH is the unique $\Sigma _1^2 $ -statement (...)
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  13.  6
    1\ Sets of Reals.J. Bagaria & W. H. Woodin - 1997 - Journal of Symbolic Logic 62 (4):1379-1428.
  14.  50
    The Cardinals Below [Ω1]<Ω1.W. Hugh Woodin - 2006 - Annals of Pure and Applied Logic 140 (1-3):161-232.
    The results of this paper concern the effective cardinal structure of the subsets of [ω1]<ω1, the set of all countable subsets of ω1. The main results include dichotomy theorems and theorems which show that the effective cardinal structure is complicated.
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  15.  18
    Sets of Reals.Joan Bagaria & W. Hugh Woodin - 1997 - Journal of Symbolic Logic 62 (4):1379-1428.
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  16.  9
    Determinacy and Jónsson Cardinals in L.S. Jackson, R. Ketchersid, F. Schlutzenberg & W. H. Woodin - 2014 - Journal of Symbolic Logic 79 (4):1184-1198.
    Assume ZF + AD +V=L and letκ< Θ be an uncountable cardinal. We show thatκis Jónsson, and that if cof = ω thenκis Rowbottom. We also establish some other partition properties.
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  17.  27
    Complexity of Reals in Inner Models of Set Theory.Boban Velickovic & W. Hugh Woodin - 1998 - Annals of Pure and Applied Logic 92 (3):283-295.
    We consider the possible complexity of the set of reals belonging to an inner model M of set theory. We show that if this set is analytic then either 1M is countable or else all reals are in M. We also show that if an inner model contains a superperfect set of reals as a subset then it contains all reals. On the other hand, it is possible to have an inner model M whose reals are an uncountable Fσ set (...)
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  18.  25
    The Jensen Covering Property.E. Schimmerling & W. H. Woodin - 2001 - Journal of Symbolic Logic 66 (4):1505-1523.
  19.  83
    Extending Partial Orders to Dense Linear Orders.Theodore A. Slaman & W. Hugh Woodin - 1998 - Annals of Pure and Applied Logic 94 (1-3):253-261.
    J. Łoś raised the following question: Under what conditions can a countable partially ordered set be extended to a dense linear order merely by adding instances of comparability ? We show that having such an extension is a Σ 1 l -complete property and so there is no Borel answer to Łoś's question. Additionally, we show that there is a natural Π 1 l -norm on the partial orders which cannot be so extended and calculate some natural ranks in that (...)
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  20. Infinity: New Research Frontiers.Michał Heller & W. H. Woodin (eds.) - 2011 - Cambridge University Press.
    Machine generated contents note: Introduction Rudy Rucker; Part I. Perspectives on Infinity from History: 1. Infinity as a transformative concept in science and theology Wolfgang Achtner; Part II. Perspectives on Infinity from Mathematics: 2. The mathematical infinity Enrico Bombieri; 3. Warning signs of a possible collapse of contemporary mathematics Edward Nelson; Part III. Technical Perspectives on Infinity from Advanced Mathematics: 4. The realm of the infinite W. Hugh Woodin; 5. A potential subtlety concerning the distinction between determinism and nondeterminism W. (...)
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  21. Infinity: New Research Frontiers.Rudy Rucker, Wolfgang Achtner, Enrico Bombieri, Edward Nelson, W. Hugh Woodin & Harvey M. Friedman - 2011 - Cambridge University Press.
    This interdisciplinary study of infinity explores the concept through the prism of mathematics and then offers more expansive investigations in areas beyond mathematical boundaries to reflect the broader, deeper implications of infinity for human intellectual thought. More than a dozen world-renowned researchers in the fields of mathematics, physics, cosmology, philosophy, and theology offer a rich intellectual exchange among various current viewpoints, rather than a static picture of accepted views on infinity.The book starts with a historical examination of the transformation of (...)
     
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  22.  32
    The Realm of the Infinite.W. Hugh Woodin - 2011 - In Michał Heller & W. H. Woodin (eds.), Infinity: New Research Frontiers. Cambridge University Press. pp. 89.
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  23.  48
    The Universe Constructed From a Sequence of Ordinals.W. Hugh Woodin - 1996 - Archive for Mathematical Logic 35 (5-6):371-383.
    We prove that if $V = L [s]$ where $s$ is an $\omega$ -sequence of ordinals then the GCH holds.
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  24.  16
    A Strong Boundedness Theorem for Dilators.A. S. Kechris & W. H. Woodin - 1991 - Annals of Pure and Applied Logic 52 (1-2):93-97.
    We prove a strong boundedness theorem for dilators: if A ⊆ DIL is Σ 1 1 , then there is a recursive dilator D 0 such that ∀ D ∈ A.
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  25.  4
    An Introduction to Independence for Analysts.H. G. Dales & W. H. Woodin - 1990 - Journal of Symbolic Logic 55 (1):361-362.
  26.  4
    The Borel Conjecture.Haim Judah, Saharon Shelah & W. H. Woodin - 1990 - Annals of Pure and Applied Logic 50 (3):255-269.
    We show the Borel Conjecture is consistent with the continuum large.
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  27.  37
    The Weak Ultrafilter Axiom.W. Hugh Woodin - 2016 - Archive for Mathematical Logic 55 (1-2):319-351.
    The main theorem is that the Ultrafilter Axiom of Woodin :115–37, 2011) must fail at all cardinals where the Axiom I0 holds, in all non-strategic extender models subject only to fairly general requirements on the non-strategic extender model.
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  28.  47
    A Potential Subtlety Concerning the Distinction Between Determinism and Nondeterminism.W. Hugh Woodin - 2011 - In Michał Heller & W. H. Woodin (eds.), Infinity: New Research Frontiers. Cambridge University Press. pp. 119.
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  29.  4
    P-Points In Models.Q. Feng & W. H. Woodin - 2003 - Annals of Pure and Applied Logic 119 (1-3):121-190.
    We show how to get canonical models from in which the nonstationary ideal on ω1 is ω1 dense and there is no P-point.
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  30.  36
    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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  31.  4
    P-Points in Qmax Models.Q. Feng & W. H. Woodin - 2003 - Annals of Pure and Applied Logic 119 (1-3):121-190.
    We show how to get canonical models from in which the nonstationary ideal on ω1 is ω1 dense and there is no P-point.
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  32.  23
    Atlanta Marriott Marquis, Atlanta, Georgia January 7–8, 2005.Matthias Aschenbrenner, Alexander Berenstein, Andres Caicedo, Joseph Mileti, Bjorn Poonen, W. Hugh Woodin & Akihiro Kanamori - 2005 - Bulletin of Symbolic Logic 11 (3).
  33.  1
    The Jensen Covering Property. E. Schimmerling & W. H. Woodin - 2001 - Journal of Symbolic Logic 66 (4):1505-1523.
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  34.  8
    Ad and the Uniqueness of the Supercompact Measures on Pω 1.W. Hugh Woodin, A. S. Kechris, D. A. Martin, Y. N. Moschavokis & Alexander S. Kechris - 1992 - Journal of Symbolic Logic 57 (1):259-261.
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  35.  11
    Super-Real Fields. Totally Ordered Fields with Additional Structure.H. Garth Dales & W. Hugh Woodin - 2000 - Bulletin of Symbolic Logic 6 (2):218-221.
  36. The Cardinals Below Vertical Bar [Omega (1)].W. H. Woodin - 2006 - Annals of Pure and Applied Logic 140 (1-3):161-232.
  37. 2002 Annual Meeting of the Association for Symbolic Logic.W. Hugh Woodin & Z. Beyond - 2003 - Bulletin of Symbolic Logic 9 (1):51.