Results for 'Carl Jockusch Jr'

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  1. 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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  2.  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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  3. 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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  4. Free Sets and Reverse Mathematics.Carl G. Jockusch Jr - 2005 - In Stephen Simpson (ed.), Reverse Mathematics 2001. pp. 104.
  5.  30
    A Minimal Pair of Π0 1 Classes.Carl G. Jockusch Jr & Robert I. Soare - 1971 - Journal of Symbolic Logic 36 (1):66 - 78.
  6. REVIEWS-Defining the Turing Jump.R. Shore, T. Slaman & Carl G. Jockusch Jr - 2001 - Bulletin of Symbolic Logic 7 (1):73-74.
  7.  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.
  8. Downey, R., Fiiredi, Z., Jockusch Jr., CG and Ruhel, LA.W. I. Gasarch, A. C. Y. Lee, M. Groszek, T. Hummel, V. S. Harizanov, H. Ishihara, B. Khoussainov, A. Nerode, I. Kalantari & L. Welch - 1998 - Annals of Pure and Applied Logic 93:263.
     
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  9.  6
    Corrigendum To: “On the Strength of Ramsey's Theorem for Pairs”.Peter Cholak, Jr} {Jockusch & Theodore A. Slaman - 2009 - Journal of Symbolic Logic 74 (4):1438-1439.
  10.  33
    Ramsey's Theorem and Recursion Theory.Carl G. Jockusch - 1972 - Journal of Symbolic Logic 37 (2):268-280.
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  11.  42
    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.
  12.  21
    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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  13.  14
    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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  14. Uniformly Introreducible Sets.Carl G. Jockusch - 1968 - Journal of Symbolic Logic 33 (4):521-536.
  15.  23
    Double Jumps of Minimal Degrees.Carl G. Jockusch & David B. Posner - 1978 - Journal of Symbolic Logic 43 (4):715 - 724.
  16. 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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  17.  8
    A Degree-Theoretic Definition of the Ramified Analytical Hierarchy.Carl G. Jockusch & Stephen G. Simpson - 1976 - Annals of Mathematical Logic 10 (1):1-32.
  18.  24
    The Degrees of Bi-Immune Sets.Carl G. Jockusch - 1969 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 15 (7-12):135-140.
  19.  7
    The Degrees of Bi‐Immune Sets.Carl G. Jockusch - 1969 - Mathematical Logic Quarterly 15 (7‐12):135-140.
  20.  22
    Correction to “a Cohesive Set Which is Not High”.Carl Jockusch & Frank Stephan - 1997 - Mathematical Logic Quarterly 43 (4):569-569.
  21. [Omnibus Review].Carl Jockusch - 1990 - Journal of Symbolic Logic 55 (1):358-360.
  22.  11
    Diagonally Non-Computable Functions and Bi-Immunity.Carl G. Jockusch & Andrew E. M. Lewis - 2013 - Journal of Symbolic Logic 78 (3):977-988.
  23.  1
    On Notions of Computability-Theoretic Reduction Between Π21 Principles.Denis R. Hirschfeldt & Carl G. Jockusch - 2016 - Journal of Mathematical Logic 16 (1):1650002.
    Several notions of computability-theoretic reducibility between [Formula: see text] principles have been studied. This paper contributes to the program of analyzing the behavior of versions of Ramsey’s Theorem and related principles under these notions. Among other results, we show that for each [Formula: see text], there is an instance of RT[Formula: see text] all of whose solutions have PA degree over [Formula: see text] and use this to show that König’s Lemma lies strictly between RT[Formula: see text] and RT[Formula: see (...)
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  24.  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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  25. The Degrees of Hyperhyperimmune Sets.Carl G. Jockusch - 1969 - Journal of Symbolic Logic 34 (3):489-493.
  26. 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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  27.  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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  28.  36
    Encodability of Kleene's O.Carl G. Jockusch & Robert I. Soare - 1973 - Journal of Symbolic Logic 38 (3):437 - 440.
  29.  14
    Π10 Classes and Boolean Combinations of Recursively Enumerable Sets.Carl G. Jockusch - 1974 - Journal of Symbolic Logic 39 (1):95-96.
  30. In Memoriam: Joseph R. Shoenfield 1927–2000.Carl G. Jockusch - 2001 - Bulletin of Symbolic Logic 7 (3):393-396.
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  31.  9
    Corrigendum To: "On the Strength of Ramsey's Theorem for Pairs".Peter A. Cholak, Carl G. Jockusch & Theodore A. Slaman - 2009 - Journal of Symbolic Logic 74 (4):1438 - 1439.
  32.  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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  33.  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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  34.  19
    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 (...)
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  35.  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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  36.  15
    Variables Affecting Sensitivity of the Human Skin to Mechanical Vibration.Carl E. Sherrick Jr - 1953 - Journal of Experimental Psychology 45 (5):273.
  37.  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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  38.  11
    An Application of Σ40 Determinacy to the Degrees of Unsolvability.Carl G. Jockusch - 1973 - Journal of Symbolic Logic 38 (2):293-294.
  39.  5
    Upward Closure of Bi‐Immune Degrees.Carl G. Jockusch - 1972 - Mathematical Logic Quarterly 18 (16‐18):285-287.
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  40.  19
    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.
  41.  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.
  42.  9
    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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  43.  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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  44.  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.
  45.  12
    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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  46.  21
    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.
  47.  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.
  48.  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.
  49.  4
    Completely Autoreducible Degrees.Carl G. Jockusch & Michael S. Paterson - 1976 - Mathematical Logic Quarterly 22 (1):571-575.
  50.  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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