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Carl G. Jockusch [48]Carl Jockusch [12]Carl G. Jockusch Jr [5]Carl Jockusch Jr [2]
  1. 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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  2.  13
    A Cohesive Set Which is Not High.Carl Jockusch & Frank Stephan - 1993 - Mathematical Logic Quarterly 39 (1):515-530.
    We study the degrees of unsolvability of sets which are cohesive . We answer a question raised by the first author in 1972 by showing that there is a cohesive set A whose degree a satisfies a' = 0″ and hence is not high. We characterize the jumps of the degrees of r-cohesive sets, and we show that the degrees of r-cohesive sets coincide with those of the cohesive sets. We obtain analogous results for strongly hyperimmune and strongly hyperhyperimmune sets (...)
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  3.  10
    Asymptotic Density and Computably Enumerable Sets.Rodney G. Downey, Carl G. Jockusch & Paul E. Schupp - 2013 - Journal of Mathematical Logic 13 (2):1350005.
  4.  15
    Richard A. Shore. Determining Automorphisms of the Recursively Enumerable Sets. Proceedings of the American Mathematical Society, Vol. 65 , Pp. 318– 325. - Richard A. Shore. The Homogeneity Conjecture. Proceedings of the National Academy of Sciences of the United States of America, Vol. 76 , Pp. 4218– 4219. - Richard A. Shore. On Homogeneity and Definability in the First-Order Theory of the Turing Degrees. The Journal of Symbolic Logic, Vol. 47 , Pp. 8– 16. - Richard A. Shore. The Arithmetic and Turing Degrees Are Not Elementarily Equivalent. Archiv Für Mathematische Logik Und Grundlagenforschung, Vol. 24 , Pp. 137– 139. - Richard A. Shore. The Structure of the Degrees of Unsolvabitity. Recursion Theory, Edited by Anil Nerode and Richard A. Shore, Proceedings of Symposia in Pure Mathematics, Vol. 42, American Mathematical Society, Providence1985, Pp. 33– 51. - Theodore A. Slaman and W. Hugh Woodin. Definability in the Turing Degrees. Illinois Journal of Mathematics, Vol. 30 , Pp. 320–. [REVIEW]Carl Jockusch - 1990 - Journal of Symbolic Logic 55 (1):358-360.
  5.  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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  6.  3
    Diagonally Non-Computable Functions and Bi-Immunity.Carl G. Jockusch & Andrew E. M. Lewis - 2013 - Journal of Symbolic Logic 78 (3):977-988.
  7.  13
    Π10 Classes and Boolean Combinations of Recursively Enumerable Sets.Carl G. Jockusch - 1974 - Journal of Symbolic Logic 39 (1):95-96.
  8.  13
    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.
  9.  7
    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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  10. 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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  11.  33
    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 2-coloring (...)
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  12.  26
    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.
  13.  7
    Asymptotic Density and the Ershov Hierarchy.Rod Downey, Carl Jockusch, Timothy H. McNicholl & Paul Schupp - 2015 - Mathematical Logic Quarterly 61 (3):189-195.
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  14. 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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  15.  28
    Ramsey's Theorem and Recursion Theory.Carl G. Jockusch - 1972 - Journal of Symbolic Logic 37 (2):268-280.
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  16.  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.
  17.  9
    An Application of Σ40 Determinacy to the Degrees of Unsolvability.Carl G. Jockusch - 1973 - Journal of Symbolic Logic 38 (2):293-294.
  18. The Degrees of Hyperhyperimmune Sets.Carl G. Jockusch - 1969 - Journal of Symbolic Logic 34 (3):489-493.
  19.  13
    Countable Thin Π01 Classes.Douglas Cenzer, Rodney Downey, Carl Jockusch & Richard A. Shore - 1993 - Annals of Pure and Applied Logic 59 (2):79-139.
    Cenzer, D., R. Downey, C. Jockusch and R.A. Shore, Countable thin Π01 classes, Annals of Pure and Applied Logic 59 79–139. A Π01 class P {0, 1}ω is thin if every Π01 subclass of P is the intersection of P with some clopen set. Countable thin Π01 classes are constructed having arbitrary recursive Cantor- Bendixson rank. A thin Π01 class P is constructed with a unique nonisolated point A and furthermore A is of degree 0’. It is shown that no (...)
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  20.  16
    Correction to “a Cohesive Set Which is Not High”.Carl Jockusch & Frank Stephan - 1997 - Mathematical Logic Quarterly 43 (4):569-569.
  21. Uniformly Introreducible Sets.Carl G. Jockusch - 1968 - Journal of Symbolic Logic 33 (4):521-536.
  22.  12
    Double Jumps of Minimal Degrees.Carl G. Jockusch & David B. Posner - 1978 - Journal of Symbolic Logic 43 (4):715 - 724.
  23.  20
    The Degrees of Bi-Immune Sets.Carl G. Jockusch - 1969 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 15 (7-12):135-140.
  24.  4
    The Degrees of Bi‐Immune Sets.Carl G. Jockusch - 1969 - Mathematical Logic Quarterly 15 (7‐12):135-140.
  25. In Memoriam: Joseph R. Shoenfield 1927–2000.Carl G. Jockusch - 2001 - Bulletin of Symbolic Logic 7 (3):393-396.
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  26.  17
    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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  27.  36
    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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  28.  29
    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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  29.  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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  30.  31
    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.
  31. [Omnibus Review].Carl Jockusch - 1990 - Journal of Symbolic Logic 55 (1):358-360.
  32.  8
    $\Pi _{1}^{0}$ Classes and Strong Degree Spectra of Relations.John Chisholm, Jennifer Chubb, Valentina S. Harizanov, Denis R. Hirschfeldt, Carl G. Jockusch, Timothy McNicholl & Sarah Pingrey - 2007 - Journal of Symbolic Logic 72 (3):1003 - 1018.
    We study the weak truth-table and truth-table degrees of the images of subsets of computable structures under isomorphisms between computable structures. In particular, we show that there is a low c.e. set that is not weak truth-table reducible to any initial segment of any scattered computable linear ordering. Countable $\Pi _{1}^{0}$ subsets of 2ω and Kolmogorov complexity play a major role in the proof.
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  33.  7
    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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  34.  15
    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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  35.  4
    Upward Closure of Bi‐Immune Degrees.Carl G. Jockusch - 1972 - Mathematical Logic Quarterly 18 (16‐18):285-287.
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  36.  16
    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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  37.  23
    Encodability of Kleene's O.Carl G. Jockusch & Robert I. Soare - 1973 - Journal of Symbolic Logic 38 (3):437 - 440.
  38.  29
    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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  39.  7
    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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  40.  17
    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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  41.  14
    Annual Meeting of the Association for Symbolic Logic, New York City, December 1987.Nicholas Goodman, Harold T. Hodes, Carl G. Jockusch & Kenneth McAloon - 1988 - Journal of Symbolic Logic 53 (4):1287-1299.
  42.  4
    Completely Autoreducible Degrees.Carl G. Jockusch & Michael S. Paterson - 1976 - Mathematical Logic Quarterly 22 (1):571-575.
  43.  14
    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.
  44.  16
    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.
  45.  16
    Ramsey's Theorem for Computably Enumerable Colorings.Tamara J. Hummel & Carl G. Jockusch - 2001 - Journal of Symbolic Logic 66 (2):873-880.
    It is shown that for each computably enumerable set P of n-element subsets of ω there is an infinite Π 0 n set $A \subseteq \omega$ such that either all n-element subsets of A are in P or no n-element subsets of A are in P. An analogous result is obtained with the requirement that A be Π 0 n replaced by the requirement that the jump of A be computable from 0 (n) . These results are best possible in (...)
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  46.  24
    A Minimal Pair of Π0 1 Classes.Carl G. Jockusch Jr & Robert I. Soare - 1971 - Journal of Symbolic Logic 36 (1):66 - 78.
  47.  5
    A Minimal Pair of Π1 0 Classes.Carl G. Jockusch & Robert I. Soare - 1971 - Journal of Symbolic Logic 36 (1):66-78.
  48.  7
    Difference Sets and Computability Theory.Rod Downey, Zoltán Füredi, Carl G. Jockusch & Lee A. Rubel - 1998 - Annals of Pure and Applied Logic 93 (1-3):63-72.
    For a set A of non-negative integers, let D be the set of non-negative differences of elements of A. Clearly, if A is computable, then D is computably enumerable. We show that every simple set which contains 0 is the difference set of some computable set and that every computably enumerable set is computably isomorphic to the difference set of some computable set. Also, we prove that there is a computable set which is the difference set of the complement of (...)
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  49.  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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  50.  12
    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.
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