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Kleene’s Amazing Second Recursion Theorem (Extended Abstract)
"... This little gem is stated unbilled and proved (completely) in the last two lines of §2 of the short note Kleene (1938). In modern notation, with all the hypotheses stated explicitly and in a strong form, it reads as follows: Theorem 1 (SRT). Fix a set V ⊆ N, and suppose that for each natural number ..."
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This little gem is stated unbilled and proved (completely) in the last two lines of §2 of the short note Kleene (1938). In modern notation, with all the hypotheses stated explicitly and in a strong form, it reads as follows: Theorem 1 (SRT). Fix a set V ⊆ N, and suppose that for each natural number n ∈ N = {0, 1, 2,...}, ϕ n: N n+1 ⇀ V is a recursive partial function of (n + 1) arguments with values in V so that the standard assumptions (1) and (2) hold with {e}(⃗x) = ϕ n e (⃗x) = ϕ n (e, ⃗x) (⃗x = (x1,..., xn) ∈ N n). (1) Every nary recursive partial function with values in V is ϕ n e for some e. (2) For all m, n, there is a recursive (total) function S = S m n: N m+1 → N such that {S(e, ⃗y)}(⃗x) = {e}(⃗y, ⃗x) (e ∈ N, ⃗y ∈ N m, ⃗x ∈ N n). Then, for every recursive, partial function f(e, ⃗y, ⃗x) of (1+m+n) arguments with values in V, there is a total recursive function ˜z(⃗y) of m arguments such that
Sets which do not have subsets of every higher degree
 Journal of Symbolic Logic
, 1978
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JSTOR is a notforprofit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. Association for Symbolic Logic is collaborating with JSTOR to digitize, preserve and extend access to The
HyperPolynomial Hierarchies and the NPJump
"... Assuming that the polynomial hierarchy (PH) does not collapse, we show the existence of ascending sequences of ptime Turing degrees of length ! CK 1 all of which are in PSPACE and uniformly hard for PH, such that successors are NPjumps of their predecessors. This is analgous to the hyperarithm ..."
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Assuming that the polynomial hierarchy (PH) does not collapse, we show the existence of ascending sequences of ptime Turing degrees of length ! CK 1 all of which are in PSPACE and uniformly hard for PH, such that successors are NPjumps of their predecessors. This is analgous to the hyperarithmetic hierarchy, which is defined similarly but with the (recursive) Turing degrees. The lack of uniform least upper bounds for ascending sequences of ptime degrees causes (the limit levels of) our hyperpolynomial hierarchy to be inherently noncanonical. This problem is investigated in depth, and various possible structures for hyperpolynomial hierarchies are explicated, as are properties of the NPjump operator on the languages which are in PSPACE but not in PH. Computer Science Department, University of Southern Maine, Portland, ME 04104. Email: fenner@cs.usm.maine.edu. Supported in part by the NSF under grants CCR 9209833 and CCR 9501794. y Computer Science Department, Bost...
SelfMonitoring Machines and an
"... this paper; thus, for any : #  CK 1 , if a (univalent) system of notations S gives a notation for :, then it will make sense to say the notation for :," and it will be denoted : S . For the remainder of the paper every mention of a system of notations will be assumed to be univalent. Also, the ..."
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this paper; thus, for any : #  CK 1 , if a (univalent) system of notations S gives a notation for :, then it will make sense to say the notation for :," and it will be denoted : S . For the remainder of the paper every mention of a system of notations will be assumed to be univalent. Also, the subscripts S will be dropped from the notation when it is clear. In addition to the three ordinal" partial recursive functions defined above it is useful to define functions able to break :=;+n into a limit ordinal notation  ; and a natural number n. For example, from +n obtain a notation for  and the natural number n. L will denote the limit part, and N the natural number part