Works by Carl G. Jockusch ( view other items matching Carl G. Jockusch, view all matches ) Disambiguations: Carl G. Jockusch Jr [29]Carl G. Jockusch [3]

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2. Carl G. Jockusch Jr & Tamara J. Lakins (2002). Generalized R-Cohesiveness and the Arithmetical Hierarchy: A Correction to "Generalized Cohesiveness". 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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3. Carl G. Jockusch & Tamara J. Lakins (2002). Generalized R-Cohesiveness and the Arithmetical Hierarchy: A Correction to "Generalized Cohesiveness". 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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4. Peter A. Cholak, Carl G. Jockusch & Theodore A. Slaman (2001). On the Strength of Ramsey's Theorem for Pairs. 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 $^n_{ denote (∀ k)RT n k . Our main result on computability is: For any n ≥ 2 and any computable (recursive) k-coloring of the n-element sets of natural numbers, there is an infinite homogeneous set X with X'' ≤ T 0 (n) . Let IΣ n and BΣ (...) Direct download (3 more) My bibliography Export citation 5. Tamara J. Hummel & Carl G. Jockusch Jr (2001). Ramsey's Theorem for Computably Enumerable Colorings. Journal of Symbolic Logic 66 (2):873-880. It is shown that for each computably enumerable set P of n-element subsets of ω there is an infinite Π 0 n set$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 (...) Direct download (3 more) My bibliography Export citation 6. Carl G. Jockusch Jr (2001). In Memoriam: Joseph R. Shoenfield, 1927-2000. Bulletin of Symbolic Logic 7 (3):393-396. Direct download (3 more) My bibliography Export citation 7. Rod Downey & Carl G. Jockusch Jr (1999). Effective Presentability of Boolean Algebras of Cantor-Bendixson Rank. Journal of Symbolic Logic 64 (1):45-52. We show that there is a computable Boolean algebra B and a computably enumerable ideal I of B such that the quotient algebra B/I is of Cantor-Bendixson rank 1 and is not isomorphic to any computable Boolean algebra. This extends a result of L. Feiner and is deduced from Feiner's result even though Feiner's construction yields a Boolean algebra of infinite Cantor-Bendixson rank. Direct download (3 more) My bibliography Export citation 8. Tamara Hummel & Carl G. Jockusch Jr (1999). Generalized Cohesiveness. 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 (...) Direct download (3 more) My bibliography Export citation 9. Carl G. Jockusch Jr & Alexandra Shlapentokh (1995). Weak Presentations of Computable Fields. 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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10. Carl G. Jockusch Jr & Robert I. Soare (1994). Boolean Algebras, Stone Spaces, and the Iterated Turing Jump. 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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11. Carl G. Jockusch Jr & Theodore A. Slaman (1993). On the Σ2-Theory of the Upper Semilattice of Turing Degrees. Journal of Symbolic Logic 58 (1):193 - 204.

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12. Carl G. Jockusch Jr & James C. Owings Jr (1990). Weakly Semirecursive Sets. 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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13. Nicholas Goodman, Harold T. Hodes, Carl G. Jockusch Jr & Kenneth McAloon (1988). Annual Meeting of the Association for Symbolic Logic, New York City, December 1987. Journal of Symbolic Logic 53 (4):1287 - 1299.

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14. Carl G. Jockusch Jr, Richard Laver, Donald Monk, Jan Mycielski & Jon Pearce (1984). Annual Meeting of the Association for Symbolic Logic: Denver, 1983. Journal of Symbolic Logic 49 (2):674 - 682.

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15. Carl G. Jockusch Jr & Richard A. Shore (1984). Pseudo-Jump Operators. II: Transfinite Iterations, Hierarchies and Minimal Covers. Journal of Symbolic Logic 49 (4):1205 - 1236.
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16. Carl G. Jockusch Jr & David B. Posner (1978). Double Jumps of Minimal Degrees. Journal of Symbolic Logic 43 (4):715 - 724.

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17. Carl G. Jockusch Jr, Robert I. Soare, William Tait & Gaisi Takeuti (1978). Meeting of the Association for Symbolic Logic: Chicago, 1977. Journal of Symbolic Logic 43 (3):614 - 619.

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18. Carl G. Jockusch Jr (1974). Π01 Classes and Boolean Combinations of Recursively Enumerable Sets. Journal of Symbolic Logic 39 (1):95 - 96.

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19. Carl G. Jockusch Jr (1973). An Application of ∑04 Determinancy to the Degrees of Unsolvability. Journal of Symbolic Logic 38 (2):293 - 294.

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20. Carl G. Jockusch Jr & Robert I. Soare (1973). Encodability of Kleene's O. Journal of Symbolic Logic 38 (3):437 - 440.

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21. Carl G. Jockusch Jr & Robert I. Soare (1973). Post's Problem and His Hypersimple Set. Journal of Symbolic Logic 38 (3):446 - 452.

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22. Carl G. Jockusch Jr (1972). Ramsey's Theorem and Recursion Theory. Journal of Symbolic Logic 37 (2):268-280.

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23. Carl G. Jockusch Jr, Joseph S. Ullian & Robert B. Barrett Jr (1972). Meeting of the Association for Symbolic Logic: St. Louis 1972. Journal of Symbolic Logic 37 (4):775-782.

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24. Carl G. Jockusch Jr & Robert I. Soare (1971). A Minimal Pair of Π0 1 Classes. Journal of Symbolic Logic 36 (1):66 - 78.

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25. Carl G. Jockusch Jr (1969). The Degrees of Hyperhyperimmune Sets. Journal of Symbolic Logic 34 (3):489-493.