We give a survey of the study of nonstandard models in recursion theory and reverse mathematics. We discuss the key notions and techniques in effective computability in nonstandard models, and their applications to problems concerning combinatorial principles in subsystems of second order arithmetic. Particular attention is given to principles related to Ramsey’s Theorem for Pairs.

This paper is essentially the author's Gödel Lecture at the ASL Logic Colloquium '09 in Sofia extended and supplemented by material from some other papers. After a brief description of traditional reverse mathematics, a computational approach to is presented. There are then discussions of some interactions between reverse mathematics and the major branches of mathematical logic in terms of the techniques they supply as well as theorems for analysis. The emphasis here is on ones that lie outside the usual main (...) systems of reverse mathematics. While retaining the usual base theory and working still within second order arithmetic, theorems are described that range from those far below the usual systems to ones far above. (shrink)

R. M. Friedberg demonstrated the existence of a recursive functional that agrees with no Banach-Mazur functional on the class of recursive functions. In this paper Friedberg's result is generalized to both α-recursive functionals and weak α-recursive functionals for all admissible ordinals α such that $\lambda , where α * is the Σ 1 -projectum of α and λ is the Σ 2 -cofinality of α. The theorem is also established for the metarecursive case, α = ω 1 , where α (...) * = λ = ω. (shrink)

If one regards an ordinal number as a generalization of a counting number, then it is natural to begin thinking in terms of computations on sets of ordinal numbers. This is precisely what Takeuti [22] had in mind when he initiated the study of recursive functions on ordinals. Kreisel and Sacks [9] too developed an ordinal recursion theory, called metarecursion theory, which specialized to the initial segment of the ordinals bounded by.The notion of admissibility was introduced by Kripke [11] and (...) Platek [14] and served to generalize metarecursion theory. Kripke called ordinal α admissible if it satisfied certain closure properties of infinitary computations. It was shown that admissibility could be equivalently formulated in terms of the replacement schema of ZF set theory and that α =is an admissible ordinal. The study of a recursion theory on an initial segment of the ordinals bounded by some arbitrary admissible α became known as α-recursion theory.Kripke [10] employed a Gödel numbering scheme to perform an arithmetiza-tion of α -recursion theory and created an analogue to Kleene'sT-predicate of ordinary recursion theory. TheT-predicate then served as the basis for showing that analogues of the major results of unrelativized o.r.t. held in α-recursion theory; namely, the α-Enumeration Theorem,T Theorem, α-Recursion Theorem, and α-Universal Function Theorem. (shrink)

A set A is symmetric enumeration (se-) reducible to a set B (A ≤\sb se B) if A is enumeration reducible to B and \barA is enumeration reducible to \barB. This reducibility gives rise to a degree structure (D\sb se) whose least element is the class of computable sets. We give a classification of ≤\sb se in terms of other standard reducibilities and we show that the natural embedding of the Turing degrees (D\sb T) into the enumeration degrees (D\sb e) (...) translates to an embedding (ι\sb se) into D\sb se that preserves least element, suprema, and infima. We define a weak and a strong jump and we observe that ι\sb se preserves the jump operator relative to the latter definition. We prove various (global) results concerning branching, exact pairs, minimal covers, and diamond embeddings in D\sb se. We show that certain classes of se-degrees are first-order definable, in particular, the classes of semirecursive, Σ\sb n ⋃ Π\sb n, Δ\sb n (for any n \in ω), and embedded Turing degrees. This last result allows us to conclude that the theory of D\sb se has the same 1-degree as the theory of Second-Order Arithmetic. (shrink)