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Extension of an Effectively Generated Class of Functions by Enumeration

Journal of Symbolic Logic 25 (3):279-280 (1960)

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Strictly Primitive Recursive Realizability, II. Completeness with Respect to Iterated Reflection and a Primitive Recursive $\omega$ -Rule.Zlatan Damnjanovic - 1998 - Notre Dame Journal of Formal Logic 39 (3):363-388.details The notion of strictly primitive recursive realizability is further investigated, and the realizable prenex sentences, which coincide with primitive recursive truths of classical arithmetic, are characterized as precisely those provable in transfinite progressions over a fragment of intuitionistic arithmetic. The progressions are based on uniform reflection principles of bounded complexity iterated along initial segments of a primitive recursively formulated system of notations for constructive ordinals. A semiformal system closed under a primitive recursively restricted -rule is described and proved equivalent to (...) Download Export citation Bookmark 2 citations
The mathematical work of S. C. Kleene.J. R. Shoenfield & S. C. Kleene - 1995 - Bulletin of Symbolic Logic 1 (1):8-43.details §1. The origins of recursion theory. In dedicating a book to Steve Kleene, I referred to him as the person who made recursion theory into a theory. Recursion theory was begun by Kleene's teacher at Princeton, Alonzo Church, who first defined the class of recursive functions; first maintained that this class was the class of computable functions ; and first used this fact to solve negatively some classical problems on the existence of algorithms. However, it was Kleene who, in his (...) Download Export citation Bookmark 5 citations
Degrees of Relative Provability.Mingzhong Cai - 2012 - Notre Dame Journal of Formal Logic 53 (4):479-489.details There are many classical connections between the proof-theoretic strength of systems of arithmetic and the provable totality of recursive functions. In this paper we study the provability strength of the totality of recursive functions by investigating the degree structure induced by the relative provability order of recursive algorithms. We prove several results about this proof-theoretic degree structure using recursion-theoretic techniques such as diagonalization and the Recursion Theorem. Download Export citation Bookmark 3 citations
On bimodal logics of provability.Lev D. Beklemishev - 1994 - Annals of Pure and Applied Logic 68 (2):115-159.details We investigate the bimodal logics sound and complete under the interpretation of modal operators as the provability predicates in certain natural pairs of arithmetical theories . Carlson characterized the provability logic for essentially reflexive extensions of theories, i.e. for pairs similar to . Here we study pairs of theories such that the gap between and is not so wide. In view of some general results concerning the problem of classification of the bimodal provability logics we are particularly interested in such (...) Download Export citation Bookmark 8 citations
Constructivity and Computability in Historical and Philosophical Perspective.Jacques Dubucs & Michel Bourdeau (eds.) - 2014 - Dordrecht, Netherland: Springer.details Ranging from Alan Turing’s seminal 1936 paper to the latest work on Kolmogorov complexity and linear logic, this comprehensive new work clarifies the relationship between computability on the one hand and constructivity on the other. The authors argue that even though constructivists have largely shed Brouwer’s solipsistic attitude to logic, there remain points of disagreement to this day. Focusing on the growing pains computability experienced as it was forced to address the demands of rapidly expanding applications, the content maps the (...) Download Export citation Bookmark 1 citation

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