47 found
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  1.  10
    Borel Equivalence Relations and Classifications of Countable Models.Greg Hjorth & Alexander S. Kechris - 1996 - Annals of Pure and Applied Logic 82 (3):221-272.
    Using the theory of Borel equivalence relations we analyze the isomorphism relation on the countable models of a theory and develop a framework for measuring the complexity of possible complete invariants for isomorphism.
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  2.  13
    Knight's Model, its Automorphism Group, and Characterizing the Uncountable Cardinals.Greg Hjorth - 2002 - Journal of Mathematical Logic 2 (01):113-144.
    We show that every ℵα can be characterized by the Scott sentence of some countable model; moreover there is a countable structure whose Scott sentence characterizes ℵ1 but whose automorphism group fails the topological Vaught conjecture on analytic sets. We obtain some partial information on Ulm type dichotomy theorems for the automorphism group of Knight's model.
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  3.  9
    Borel Equivalence Relations Induced by Actions of the Symmetric Group.Greg Hjorth, Alexander S. Kechris & Alain Louveau - 1998 - Annals of Pure and Applied Logic 92 (1):63-112.
    We consider Borel equivalence relations E induced by actions of the infinite symmetric group, or equivalently the isomorphism relation on classes of countable models of bounded Scott rank. We relate the descriptive complexity of the equivalence relation to the nature of its complete invariants. A typical theorem is that E is potentially Π03 iff the invariants are countable sets of reals, it is potentially Π04 iff the invariants are countable sets of countable sets of reals, and so on. The proofs (...)
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  4.  21
    New Dichotomies for Borel Equivalence Relations.Greg Hjorth & Alexander S. Kechris - 1997 - Bulletin of Symbolic Logic 3 (3):329-346.
  5.  19
    Regularity Properties for Dominating Projective Sets.Jörg Brendle, Greg Hjorth & Otmar Spinas - 1995 - Annals of Pure and Applied Logic 72 (3):291-307.
    We show that every dominating analytic set in the Baire space has a dominating closed subset. This improves a theorem of Spinas [15] saying that every dominating analytic set contains the branches of a uniform tree, i.e. a superperfect tree with the property that for every splitnode all the successor splitnodes have the same length. In [15], a subset of the Baire space is called u-regular if either it is not dominating or it contains the branches of a uniform tree, (...)
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  6. Thin Equivalence Relations and Effective Decompositions.Greg Hjorth - 1993 - Journal of Symbolic Logic 58 (4):1153-1164.
    Let E be a Σ1 1 equivalence relation for which there does not exist a perfect set of inequivalent reals. If 0# exists or if V is a forcing extension of L, then there is a good ▵1 2 well-ordering of the equivalence classes.
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  7.  27
    Analytic Equivalence Relations and Ulm-Type Classifications.Greg Hjorth & Alexander S. Kechris - 1995 - Journal of Symbolic Logic 60 (4):1273-1300.
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  8.  9
    A Note on Counterexamples to the Vaught Conjecture.Greg Hjorth - 2007 - Notre Dame Journal of Formal Logic 48 (1):49-51.
    If some infinitary sentence provides a counterexample to Vaught's Conjecture, then there is an infinitary sentence which also provides a counterexample but has no model of cardinality bigger than ℵ₁.
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  9.  30
    An Absoluteness Principle for Borel Sets.Greg Hjorth - 1998 - Journal of Symbolic Logic 63 (2):663-693.
  10.  28
    A Dichotomy for the Definable Universe.Greg Hjorth - 1995 - Journal of Symbolic Logic 60 (4):1199-1207.
    In the presence of large cardinals, or sufficient determinacy, every equivalence relation in L(R) either admits a wellordered separating family or continuously reduces E 0.
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  11. Two Applications of Inner Model Theory to the Study of $\underset \sim \to{\sigma}{}_{2}^{1}$ Sets.Greg Hjorth - 1996 - Bulletin of Symbolic Logic 2 (1):94 - 107.
  12.  11
    Strong Cardinals in the Core Model.Kai Hauser & Greg Hjorth - 1997 - Annals of Pure and Applied Logic 83 (2):165-198.
  13.  13
    Borel Equivalence Relations Which Are Highly Unfree.Greg Hjorth - 2008 - Journal of Symbolic Logic 73 (4):1271-1277.
    There is an ergodic, measure preserving, countable Borel equivalence relation E on a standard Borel probability space (X, µ) such that E\c is not essentially free on any conull C ⊂ X.
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  14.  55
    Some Applications of Coarse Inner Model Theory.Greg Hjorth - 1997 - Journal of Symbolic Logic 62 (2):337-365.
    The Martin-Steel coarse inner model theory is employed in obtaining new results in descriptive set theory. $\underset{\sim}{\Pi}$ determinacy implies that for every thin Σ 1 2 equivalence relation there is a Δ 1 3 real, N, over which every equivalence class is generic--and hence there is a good Δ 1 2 (N ♯ ) wellordering of the equivalence classes. Analogous results are obtained for Π 1 2 and Δ 1 2 quasilinear orderings and $\underset{\sim}{\Pi}^1_2$ determinacy is shown to imply that (...)
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  15.  13
    Effective Cardinals of Boldface Pointclasses.Alessandro Andretta, Greg Hjorth & Itay Neeman - 2007 - Journal of Mathematical Logic 7 (1):35-82.
    Assuming AD + DC, we characterize the self-dual boldface pointclasses which are strictly larger than the pointclasses contained in them: these are exactly the clopen sets, the collections of all sets of Wadge rank [Formula: see text], and those of Wadge rank [Formula: see text] when ξ is limit.
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  16.  9
    The Classification Problem for P-Local Torsion-Free Abelian Groups of Rank Two.Greg Hjorth & Simon Thomas - 2006 - Journal of Mathematical Logic 6 (2):233-251.
    We prove that if p ≠ q are distinct primes, then the classification problems for p-local and q-local torsion-free abelian groups of rank two are incomparable with respect to Borel reducibility.
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  17.  24
    L. A. Harrington, A. S. Kechris, and A. Louveau. A Glimm–Effros Dichotomy for Borel Equivalence Relations. Journal of the American Mathematical Society, Vol. 3 , Pp. 903–928. - Alain Louveau and Boban Velickovic. A Note on Borel Equivalence Relations. Proceedings of the American Mathematical Society, Vol. 120 , Pp. 255–259. - Alexander S. Kechris and Alain Louveau. The Classification Ofhypersmooth Equivalence Relations. Journal of the American Mathematical Society, Vol. 10 , Pp. 215–242. [REVIEW]Greg Hjorth - 1998 - Journal of Symbolic Logic 63 (2):749-750.
  18. REVIEWS-Papers.M. Foreman, A. Kechris & Greg Hjorth - 2001 - Bulletin of Symbolic Logic 7 (4):545-546.
  19. REVIEWS-Set Theory.T. Jech & Greg Hjorth - 2005 - Bulletin of Symbolic Logic 11 (2):243-245.
     
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  20.  40
    Universal Sets for Pointsets Properly on the N Th Level of the Projective Hierarchy.Greg Hjorth, Leigh Humphries & Arnold W. Miller - 2013 - Journal of Symbolic Logic 78 (1):237-244.
    The Axiom of Projective Determinacy implies the existence of a universal $\utilde{\Pi}^{1}_{n}\setminus\utilde{\Delta}^{1}_{n}$ set for every $n \geq 1$. Assuming $\text{\upshape MA}(\aleph_{1})+\aleph_{1}=\aleph_{1}^{\mathbb{L}}$ there exists a universal $\utilde{\Pi}^{1}_{1}\setminus\utilde{\Delta}^{1}_{1}$ set. In ZFC there is a universal $\utilde{\Pi}^{0}_{\alpha}\setminus\utilde{\Delta}^{0}_{\alpha}$ set for every $\alpha$.
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  21. A Boundedness Lemma for Iterations.Greg Hjorth - 2001 - Journal of Symbolic Logic 66 (3):1058-1072.
  22.  22
    Glimm-Effros for Coanalytic Equivalence Relations.Greg Hjorth - 2009 - Journal of Symbolic Logic 74 (2):402-422.
    Assuming every real has a sharp, we prove that for any $\mathop \prod \limits_\~ _1^1 $ equivalence relation either Borel reduces E₀ or in a $\mathop \Delta \limits_\~ _3^1 $ manner allows the assignment of bounded subsets of ω₁ as complete invariants.
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  23.  26
    Treeable Equivalence Relations.Greg Hjorth - 2012 - Journal of Mathematical Logic 12 (1):1250003-.
    There are continuum many ≤B-incomparable equivalence relations induced by a free, Borel action of a countable non-abelian free group — and hence, there are 2α0 many treeable countable Borel equivalence relations which are incomparable in the ordering of Borel reducibility.
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  24.  20
    Matthew Foreman. A Descriptive View of Ergodic Theory. Descriptive Set Theory and Dynamical Systems, Edited by M. Foreman, A. S. Kechris, A. Louveau, and B. Weiss, London Mathematical Society Lecture Note Series, No. 277, Cambridge University Press, Cambridge, New York, Etc., 2000, Pp. 87–171. [REVIEW]Greg Hjorth - 2001 - Bulletin of Symbolic Logic 7 (4):545-546.
  25.  23
    Uniquely Undefinable Elements.Greg Hjorth - 2010 - Journal of Symbolic Logic 75 (1):269-274.
    There exists a model in a countable language having a unique element which is not definable in $\scr{L}_{\omega _{1},\omega}$.
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  26.  14
    We Announce Two New Dichotomy Theorems for Borel Equivalence Rela-Tions, and Present the Results in Context by Giving an Overview of Related Recent Developments. § 1. Introduction. For X a Polish (Ie, Separable, Completely Metrizable) Space and E a Borel Equivalence Relation on X, a (Complete) Classification. [REVIEW]Greg Hjorth & Alexander S. Kechris - 1997 - Bulletin of Symbolic Logic 3 (3).
  27.  14
    2001–2002 Winter Meeting of the Association for Symbolic Logic.Greg Hjorth - 2002 - Bulletin of Symbolic Logic 8 (2):312-318.
  28.  13
    Variations of the Martin-Solovay Tree.Greg Hjorth - 1996 - Journal of Symbolic Logic 61 (1):40-51.
    Assuming $\underset{\sim}{\Pi}^1_2$ determinacy, the model L[ T 2 ] does not depend on the choice of T 2.
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  29.  11
    Independently Axiomatizable ℒω1,Ω Theories.Greg Hjorth & Ioannis A. Souldatos - 2009 - Journal of Symbolic Logic 74 (4):1273-1286.
    In partial answer to a question posed by Arnie Miller [4] and X. Caicedo [2] we obtain sufficient conditions for an ℒω1,ω theory to have an independent axiomatization. As a consequence we obtain two corollaries: The first, assuming Vaught's Conjecture, every ℒω1,ω theory in a countable language has an independent axiomatization. The second, this time outright in ZFC, every intersection of a family of Borel sets can be formed as the intersection of a family of independent Borel sets.
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  30.  10
    Review: L. A. Harrington, A. S. Kechris, A. Louveau, A Glimm-Effros Dichotomy for Borel Equivalence Relations. [REVIEW]Greg Hjorth - 1998 - Journal of Symbolic Logic 63 (2):749-750.
  31.  10
    Review: Matthew Foreman, M. Foreman, A. S. Kechris, A. Louveau, B. Weiss, A Descriptive View of Ergodic Theory; Alexander S. Kechris, Descriptive Dynamics. [REVIEW]Greg Hjorth - 2001 - Bulletin of Symbolic Logic 7 (4):545-546.
  32.  10
    Borel Structures and Borel Theories.Greg Hjorth & André Nies - 2011 - Journal of Symbolic Logic 76 (2):461 - 476.
    We show that there is a complete, consistent Borel theory which has no "Borel model" in the following strong sense: There is no structure satisfying the theory for which the elements of the structure are equivalence classes under some Borel equivalence relation and the interpretations of the relations and function symbols are uniformly Borel. We also investigate Borel isomorphisms between Borel structures.
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  33.  7
    Descriptive Set Theory and Dynamical Systems.Greg Hjorth - 2001 - Bulletin of Symbolic Logic 7 (4):545-546.
  34.  9
    Jech T.. Set Theory. The Third Millennium Edition, Revised and Expanded. Springer-Verlag, Berlin, 2003, Viii+ 769 Pp. [REVIEW]Greg Hjorth - 2005 - Bulletin of Symbolic Logic 11 (2):243-245.
  35.  9
    On ℵ1 Many Minimal Models.Greg Hjorth - 1996 - Journal of Symbolic Logic 61 (3):906 - 919.
    The existence of a countable complete theory with exactly ℵ 1 many minimal models is independent of ZFC + ¬CH.
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  36.  7
    Some Applications of Illfoundedness.Greg Hjorth - 1996 - Archive for Mathematical Logic 35 (3):131-144.
    It is possible to completely characterize which countable models generated by 0# exist inL. This in turn has applications in the study of analytic equivalence relations; for instance, ifE is∑ 1 1 and every invariant∑ 1 1 (0#) set isΔ 1 1 , thenE has at most ℵ0 many equivalence classes.
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  37.  9
    A Lemma for Cost Attained.Greg Hjorth - 2006 - Annals of Pure and Applied Logic 143 (1-3):87-102.
    A treeable ergodic equivalence relation of integer cost is generated by a free action of the free group on the corresponding number of generators. Every countable treeable ergodic equivalence relation is induced by the free action of some countable group.
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  38.  9
    $\pi^1_1$ Wellfounded Relations.Greg Hjorth - 1994 - Notre Dame Journal of Formal Logic 35 (4):542-549.
    If there is a good $\Delta^1_3$ wellordering of the reals, then there is a $\Pi^1_1$ wellfounded relation for which the comparison relation is not projective.
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  39.  4
    Wellfounded Relations.Greg Hjorth - 1994 - Notre Dame Journal of Formal Logic 35 (4):542-549.
    If there is a good wellordering of the reals, then there is a wellfounded relation for which the comparison relation is not projective.
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  40.  13
    Π12 Wadge Degrees.Greg Hjorth - 1996 - Annals of Pure and Applied Logic 77 (1):53-74.
    Suppose that any two Π12 sets are comparable in the sense of Wadge degrees. Then every real has a dagger. This argument proceeds by using the Dodd-Jensen core model theory to show that x ε ωω along with, say, “0† implies the existence of a Π12 norm of length u2. As a result of more recent work by John Steel, the same argument will extend to show that the Wadge comparability of all Π12 sets implies Π12 determinacy.
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  41.  9
    Bi-Borel Reducibility of Essentially Countable Borel Equivalence Relations.Greg Hjorth - 2005 - Journal of Symbolic Logic 70 (3):979 - 992.
  42.  16
    A Dichotomy Theorem for Turbulence.Greg Hjorth - 2002 - Journal of Symbolic Logic 67 (4):1520-1540.
  43.  9
    < I> Π_< Sup> 1< Sub> 2 Wadge Degrees.Greg Hjorth - 1996 - Annals of Pure and Applied Logic 77 (1):53-74.
  44.  12
    Cardinalities in the Projective Hierarchy.Greg Hjorth - 2002 - Journal of Symbolic Logic 67 (4):1351-1372.
  45.  1
    On $Aleph_1$ Many Minimal Models.Greg Hjorth - 1996 - Journal of Symbolic Logic 61 (3):906-919.
    The existence of a countable complete theory with exactly $\aleph_1$ many minimal models is independent of $\mathrm{ZFC} + \neg\mathrm{CH}$.
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  46.  1
    Π12 Wadge Degrees.Greg Hjorth - 1996 - Annals of Pure and Applied Logic 77 (1):53-74.
  47.  26
    Two Applications Of Inner Model Theory To The Study Of \sigma^1_2 Sets.Greg Hjorth - 1996 - Bulletin of Symbolic Logic 2 (1):94-107.