Results for 'Carl G. Jockusch Jr'

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  1.  29
    A Minimal Pair of Π0 1 Classes.Carl G. Jockusch Jr & Robert I. Soare - 1971 - Journal of Symbolic Logic 36 (1):66 - 78.
  2. A Letter From.Carl G. Jockusch Jr - 1996 - In Piergiorgio Odifreddi (ed.), Kreiseliana. About and Around Georg Kreisel. A K Peters.
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  3. Free Sets and Reverse Mathematics.Carl G. Jockusch Jr - 2005 - In Stephen Simpson (ed.), Reverse Mathematics 2001. pp. 104.
  4. REVIEWS-Defining the Turing Jump.R. Shore, T. Slaman & Carl G. Jockusch Jr - 2001 - Bulletin of Symbolic Logic 7 (1):73-74.
  5.  34
    2001 Annual Meeting of the Association for Symbolic Logic.Joan Feigenbaum, Haim Gaifman, Jean-Yves Girard, C. Ward Henson, Denis Hirschfeldt, Carl G. Jockusch Jr, Saul Kripke, Salma Kuhlmann, John C. Mitchell & Ernest Schimmerling - 2001 - Bulletin of Symbolic Logic 7 (3):420-435.
  6.  31
    Ramsey's Theorem and Recursion Theory.Carl G. Jockusch - 1972 - Journal of Symbolic Logic 37 (2):268-280.
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  7.  39
    Pseudo-Jump Operators. II: Transfinite Iterations, Hierarchies and Minimal Covers.Carl G. Jockusch & Richard A. Shore - 1984 - Journal of Symbolic Logic 49 (4):1205 - 1236.
  8.  47
    Ramsey's Theorem and Cone Avoidance.Damir Dzhafarov & Carl Jockusch Jr - 2009 - Journal of Symbolic Logic 74 (2):557 - 578.
    It was shown by Cholak, Jockusch, and Slaman that every computable 2-coloring of pairs admits an infinite low₂ homogeneous set H. We answer a question of the same authors by showing that H may be chosen to satisfy in addition $C\,\not \leqslant _T \,H$ , where C is a given noncomputable set. This is shown by analyzing a new and simplified proof of Seetapun's cone avoidance theorem for Ramsey's theorem. We then extend the result to show that every computable (...)
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  9. Stability and Posets.Carl Jockusch Jr, Bart Kastermans, Steffen Lempp, Manuel Lerman & Reed Solomon - 2009 - Journal of Symbolic Logic 74 (2):693 - 711.
    Hirschfeldt and Shore have introduced a notion of stability for infinite posets. We define an arguably more natural notion called weak stability, and we study the existence of infinite computable or low chains or antichains, and of infinite $\Pi _1^0 $ chains and antichains, in infinite computable stable and weakly stable posets. For example, we extend a result of Hirschfeldt and Shore to show that every infinite computable weakly stable poset contains either an infinite low chain or an infinite computable (...)
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  10.  13
    Degrees of Orderings Not Isomorphic to Recursive Linear Orderings.Carl G. Jockusch & Robert I. Soare - 1991 - Annals of Pure and Applied Logic 52 (1-2):39-64.
    It is shown that for every nonzero r.e. degree c there is a linear ordering of degree c which is not isomorphic to any recursive linear ordering. It follows that there is a linear ordering of low degree which is not isomorphic to any recursive linear ordering. It is shown further that there is a linear ordering L such that L is not isomorphic to any recursive linear ordering, and L together with its ‘infinitely far apart’ relation is of low (...)
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  11.  23
    Double Jumps of Minimal Degrees.Carl G. Jockusch & David B. Posner - 1978 - Journal of Symbolic Logic 43 (4):715 - 724.
  12. Uniformly Introreducible Sets.Carl G. Jockusch - 1968 - Journal of Symbolic Logic 33 (4):521-536.
  13.  6
    A Degree-Theoretic Definition of the Ramified Analytical Hierarchy.Carl G. Jockusch & Stephen G. Simpson - 1976 - Annals of Mathematical Logic 10 (1):1-32.
  14.  23
    The Degrees of Bi-Immune Sets.Carl G. Jockusch - 1969 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 15 (7-12):135-140.
  15.  5
    The Degrees of Bi‐Immune Sets.Carl G. Jockusch - 1969 - Mathematical Logic Quarterly 15 (7‐12):135-140.
  16.  10
    Diagonally Non-Computable Functions and Bi-Immunity.Carl G. Jockusch & Andrew E. M. Lewis - 2013 - Journal of Symbolic Logic 78 (3):977-988.
  17. On the Strength of Ramsey's Theorem for Pairs.Peter A. Cholak, Carl G. Jockusch & Theodore A. Slaman - 2001 - Journal of Symbolic Logic 66 (1):1-55.
    We study the proof-theoretic strength and effective content of the infinite form of Ramsey's theorem for pairs. Let RT n k denote Ramsey's theorem for k-colorings of n-element sets, and let RT $^n_{ denote (∀ k)RT n k . Our main result on computability is: For any n ≥ 2 and any computable (recursive) k-coloring of the n-element sets of natural numbers, there is an infinite homogeneous set X with X'' ≤ T 0 (n) . Let IΣ n and BΣ (...)
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  18.  34
    Boolean Algebras, Stone Spaces, and the Iterated Turing Jump.Carl G. Jockusch & Robert I. Soare - 1994 - Journal of Symbolic Logic 59 (4):1121 - 1138.
    We show, roughly speaking, that it requires ω iterations of the Turing jump to decode nontrivial information from Boolean algebras in an isomorphism invariant fashion. More precisely, if α is a recursive ordinal, A is a countable structure with finite signature, and d is a degree, we say that A has αth-jump degree d if d is the least degree which is the αth jump of some degree c such there is an isomorphic copy of A with universe ω in (...)
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  19. The Degrees of Hyperhyperimmune Sets.Carl G. Jockusch - 1969 - Journal of Symbolic Logic 34 (3):489-493.
  20. Weak Presentations of Computable Fields.Carl G. Jockusch & Alexandra Shlapentokh - 1995 - Journal of Symbolic Logic 60 (1):199 - 208.
    It is shown that for any computable field K and any r.e. degree a there is an r.e. set A of degree a and a field F ≅ K with underlying set A such that the field operations of F (including subtraction and division) are extendible to (total) recursive functions. Further, it is shown that if a and b are r.e. degrees with b ≤ a, there is a 1-1 recursive function $f: \mathbb{Q} \rightarrow \omega$ such that f(Q) ∈ a, (...)
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  21.  14
    Π10 Classes and Boolean Combinations of Recursively Enumerable Sets.Carl G. Jockusch - 1974 - Journal of Symbolic Logic 39 (1):95-96.
  22.  34
    Encodability of Kleene's O.Carl G. Jockusch & Robert I. Soare - 1973 - Journal of Symbolic Logic 38 (3):437 - 440.
  23.  20
    Upward Closure of Bi-Immune Degrees.Carl G. Jockusch - 1972 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 18 (16-18):285-287.
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  24.  22
    Asymptotic Density and Computably Enumerable Sets.Rodney G. Downey, Carl G. Jockusch & Paul E. Schupp - 2013 - Journal of Mathematical Logic 13 (2):1350005.
    We study connections between classical asymptotic density, computability and computable enumerability. In an earlier paper, the second two authors proved that there is a computably enumerable set A of density 1 with no computable subset of density 1. In the current paper, we extend this result in three different ways: The degrees of such sets A are precisely the nonlow c.e. degrees. There is a c.e. set A of density 1 with no computable subset of nonzero density. There is a (...)
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  25.  29
    Weakly Semirecursive Sets.Carl G. Jockusch & James C. Owings - 1990 - Journal of Symbolic Logic 55 (2):637-644.
    We introduce the notion of "semi-r.e." for subsets of ω, a generalization of "semirecursive" and of "r.e.", and the notion of "weakly semirecursive", a generalization of "semi-r.e.". We show that A is weakly semirecursive iff, for any n numbers x 1 ,...,x n , knowing how many of these numbers belong to A is equivalent to knowing which of these numbers belong to A. It is shown that there exist weakly semirecursive sets that are neither semi-r.e. nor co-semi-r.e. On the (...)
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  26.  11
    An Application of Σ40 Determinacy to the Degrees of Unsolvability.Carl G. Jockusch - 1973 - Journal of Symbolic Logic 38 (2):293-294.
  27.  5
    Upward Closure of Bi‐Immune Degrees.Carl G. Jockusch - 1972 - Mathematical Logic Quarterly 18 (16‐18):285-287.
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  28.  17
    Completely Autoreducible Degrees.Carl G. Jockusch & Michael S. Paterson - 1976 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 22 (1):571-575.
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  29.  8
    Post's Problem and His Hypersimple Set.Carl G. Jockusch & Robert I. Soare - 1973 - Journal of Symbolic Logic 38 (3):446 - 452.
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  30.  4
    Completely Autoreducible Degrees.Carl G. Jockusch & Michael S. Paterson - 1976 - Mathematical Logic Quarterly 22 (1):571-575.
  31.  7
    Coarse Reducibility and Algorithmic Randomness.Denis R. Hirschfeldt, Carl G. Jockusch, Rutger Kuyper & Paul E. Schupp - 2016 - Journal of Symbolic Logic 81 (3):1028-1046.
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  32.  10
    A Minimal Pair of Π1 0 Classes.Carl G. Jockusch & Robert I. Soare - 1971 - Journal of Symbolic Logic 36 (1):66-78.
  33.  19
    Generalized Cohesiveness.Tamara Hummel & Carl G. Jockusch - 1999 - Journal of Symbolic Logic 64 (2):489-516.
    We study some generalized notions of cohesiveness which arise naturally in connection with effective versions of Ramsey's Theorem. An infinite set A of natural numbers is n-cohesive (respectively, n-r-cohesive) if A is almost homogeneous for every computably enumerable (respectively, computable) 2-coloring of the n-element sets of natural numbers. (Thus the 1-cohesive and 1-r-cohesive sets coincide with the cohesive and r-cohesive sets, respectively.) We consider the degrees of unsolvability and arithmetical definability levels of n-cohesive and n-r-cohesive sets. For example, we show (...)
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  34.  39
    Chains and Antichains in Partial Orderings.Valentina S. Harizanov, Carl G. Jockusch & Julia F. Knight - 2009 - Archive for Mathematical Logic 48 (1):39-53.
    We study the complexity of infinite chains and antichains in computable partial orderings. We show that there is a computable partial ordering which has an infinite chain but none that is ${\Sigma _{1}^{1}}$ or ${\Pi _{1}^{1}}$ , and also obtain the analogous result for antichains. On the other hand, we show that every computable partial ordering which has an infinite chain must have an infinite chain that is the difference of two ${\Pi _{1}^{1}}$ sets. Our main result is that there (...)
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  35.  15
    On Self-Embeddings of Computable Linear Orderings.Rodney G. Downey, Carl Jockusch & Joseph S. Miller - 2006 - Annals of Pure and Applied Logic 138 (1):52-76.
    The Dushnik–Miller Theorem states that every infinite countable linear ordering has a nontrivial self-embedding. We examine computability-theoretical aspects of this classical theorem.
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  36.  16
    Annual Meeting of the Association for Symbolic Logic Denver, 1983.Carl G. Jockusch, Richard Laver, Donald Monk, Jan Mycielski & Jon Pearce - 1984 - Journal of Symbolic Logic 49 (2):674 - 682.
  37.  31
    Generalized R-Cohesiveness and the Arithmetical Hierarchy: A Correction to "Generalized Cohesiveness".Carl G. Jockusch & Tamara J. Lakins - 2002 - Journal of Symbolic Logic 67 (3):1078 - 1082.
    For $X \subseteq \omega$ , let $\lbrack X \rbrack^n$ denote the class of all n-element subsets of X. An infinite set $A \subseteq \omega$ is called n-r-cohesive if for each computable function $f: \lbrack \omega \rbrack^n \rightarrow \lbrace 0, 1 \rbrace$ there is a finite set F such that f is constant on $\lbrack A - F \rbrack^n$ . We show that for each n ≥ 2 there is no $\prod_n^0$ set $A \subseteq \omega$ which is n-r-cohesive. For n = (...)
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  38. In Memoriam: Joseph R. Shoenfield 1927–2000.Carl G. Jockusch - 2001 - Bulletin of Symbolic Logic 7 (3):393-396.
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  39.  18
    Meeting of the Association for Symbolic Logic: St. Louis 1972.Carl G. Jockusch, Joseph S. Ullian & Robert B. Barrett - 1972 - Journal of Symbolic Logic 37 (4):775-782.
  40.  17
    Meeting of the Association for Symbolic Logic, Chicago, 1977.Carl G. Jockusch, Robert I. Soare, William Tait & Gaisi Takeuti - 1978 - Journal of Symbolic Logic 43 (3):614 - 619.
  41.  52
    Computability, Enumerability, Unsolvability, Directions in Recursion Theory, Edited by S. B. Cooper, T. A. Slaman, and S. S. Wainer, London Mathematical Society Lecture Note Series, No. 224, Cambridge University Press, Cambridge, New York, and Oakleigh, Victoria, 1996, Vii + 347 Pp. - Leo Harrington and Robert I. Soare, Dynamic Properties of Computably Enumerable Sets, Pp. 105–121. - Eberhard Herrmann, On the ∀∃-Theory of the Factor Lattice by the Major Subset Relation, Pp. 139–166. - Manuel Lerman, Embeddings Into the Recursively Enumerable Degrees, Pp. 185–204. - Xiaoding Yi, Extension of Embeddings on the Recursively Enumerable Degrees Modulo the Cappable Degrees, Pp. 313–331. - André Nies, Relativization of Structures Arising From Computability Theory. Pp. 219–232. - Klaus Ambos-Spies, Resource-Bounded Genericity. Pp. 1–59. - Rod Downey, Carl G. Jockusch, and Michael Stob. Array Nonrecursive Degrees and Genericity, Pp. 93–104. - Masahiro Kumabe, Degrees of Generic Sets, Pp. 167–183. [REVIEW]C. T. Chong - 1999 - Journal of Symbolic Logic 64 (3):1362-1365.
  42.  4
    Double Jumps of Minimal Degrees.Carl G. Jockusch, David B. Posner, Richard L. Epstein & Richard A. Shore - 1985 - Journal of Symbolic Logic 50 (2):550-552.
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  43.  5
    Fine Degrees of Word Problems of Cancellation Semigroups.Carl G. Jockusch - 1980 - Mathematical Logic Quarterly 26 (1‐6):93-95.
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  44.  18
    Fine Degrees of Word Problems of Cancellation Semigroups.Carl G. Jockusch - 1980 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 26 (1-6):93-95.
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  45.  2
    Lachlan A. H.. Some Notions of Reducibility and Productiveness. Zeitschrift Für Mathematische Logik Und Grundlagen der Mathematik, Vol. 11 , Pp. 17–44. [REVIEW]Carl G. Jockusch - 1970 - Journal of Symbolic Logic 35 (3):478-478.
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  46.  18
    Lerman Manuel. Degrees of Unsolvability. Local and Global Theory. Perspectives in Mathematical Logic. Springer-Verlag, Berlin, Heidelberg, New York, and Tokyo, 1983, Xiii + 307 Pp. [REVIEW]Carl G. Jockusch - 1985 - Journal of Symbolic Logic 50 (2):549-550.
  47.  5
    Mathematical Research Letters.Carl G. Jockusch - 2001 - Bulletin of Symbolic Logic 7 (1):73-75.
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  48.  3
    Review: A. H. Lachlan, Lower Bounds for Pairs of Recursively Enumerable Degrees. [REVIEW]Carl G. Jockusch - 1972 - Journal of Symbolic Logic 37 (3):611-611.
  49.  2
    Review: A. H. Lachlan, Some Notions of Reducibility and Productiveness. [REVIEW]Carl G. Jockusch - 1970 - Journal of Symbolic Logic 35 (3):478-478.
  50.  17
    Richard A. Shore and Theodore A. Slaman. Defining the Turing Jump. Mathematical Research Letters, Vol. 6 , Pp. 711–722.Carl G. Jockusch - 2001 - Bulletin of Symbolic Logic 7 (1):73-75.
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