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The next admissible set1

  • K. J. Barwise (a1), R. O. Gandy (a2) and Y. N. Moschovakis (a3)

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In this paper we describe generalizations of several approaches to the hyperarithmetic hierarchy, show how they are related to the Kripke-Platek theory of admissible ordinals and sets, and study conditions under which the various approaches remain equivalent.

To put matters in some perspective, let us first review various approaches to the theory of hyperarithmetic sets. For most purposes, it is convenient to first define the semi-hyperarithmetic (semi-HA) subsets of N. A set is then said to be hyperarithmetic (HA) if both it and its complement are semi-HA. A total number-theoretic function is HA if its graph is HA.

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Most of the results of this paper were obtained during the spring of 1968 while Barwise and Gandy were attending the UCLA Logic Year, the first as an NSF Postdoctoral Fellow, the second as Visiting Associate Professor. They record here their gratitude to the Mathematics Department of UCLA for its gracious hospitality. The research of Barwise and Moschovakis was also supported in part by NSF Grants.

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[1]Barwise, J., Infinitary logic and admissible sets, this Journal, vol. 34 (1969), pp. 226252 (referred to as ILAS).
[2]Barwise, J., Applications of strict-Π11 predicates to infinitary logic, this Journal, vol. 34 (1969), pp. 409423.
[3]Barwise, J., The Gandy-Kreisel-Tait theorem for countable sets, (in preparation).
[4]Gandy, R. O., Computable functionals of finite type, Sets, Models and Recursion Theory, North Holland, 1967, pp. 202242.
[5]Gandy, R. O., Kreisel, G. and Tait, W. W., Set existence, Bulletin de l'Académie Polonaise des Sciences, vol. 8 (1960), pp. 577583.
[6]Grzegorczyk, A., Mostowski, A. and Ryll-Nardzewski, C., The classical and the ω-complete arithmetic, this Journal, vol. 23 (1958), pp. 188206.
[7]Jensen, R. and Karp, C., Primitive recursive set functions, Proceedings of the Summer Institute on Axiomatic Set Theory, UCLA 1967, (to appear).
[8]Kleene, S. C., Hierarchies of number-theoretic predicates, Bulletin of the American Mathematical Society, vol. 61 (1955), pp. 193213.
[9]Kleene, S. C., Quantification of number theoretic functions, Compositio Mathematica, vol. 14 (1959), pp. 2341.
[10]Kleene, S. C., Recursive functionals and quantifiers of finite types, I, Transactions of the American Mathematical Society, vol. 91 (1959), pp. 152.
[11]Kreisel, G., Set theoretic problems suggested by the notion of potential totality, Infinitistic Methods, Warsaw (1961), pp. 103140.
[12]Kreisel, G., Model theoretic invariants, The Theory of Models, North Holland Publishing Co. (1965), pp. 190205.
[13]Kreisel, G., A survey of proof theory, this Journal, vol. 33 (1968), pp. 321338.
[14]Kreisel, G. and Sacks, G., Metarecursive sets, this Journal, vol. 30 (1965), pp. 318338.
[15]Kripke, S., Transfinite recursion on admissible ordinals, I, II (abstract), this Journal, vol. 29 (1964), p. 162.
[16]Kunen, K., Implicit definability and infinitary languages, this Journal, vol. 33 (1968), pp. 446451.
[17]Lorenzen, P. and Myhill, J., Constructive definitions of certain analytic sets of numbers, this Journal, vol. 24 (1959), pp. 3749.
[18]Moschovakis, Y. N., Abstract first order computability, I, II, Transactions of the American Mathematical Society, vol. 138 (1969), pp. 427464, 465–504 (referred to as AFOQ.
[19]Moschovakis, Y. N., Abstract computability and invariant definability, this Journal (to appear) (referred to as ACID).
[20]Moschovakis, Y. N., The Suslin–Kleene theorem for countable structures, Duke Journal of Mathematics, vol. 37 (1970), pp. 344352.
[21]Platek, R., Foundations of recursion theory, Ph.D. Thesis and supplement, Stanford Univ. 1966.
[22]Spector, C., Inductively defined sets of natural numbers, Infinitistic Methods, Warsaw, 1961, pp. 97102.

The next admissible set1

  • K. J. Barwise (a1), R. O. Gandy (a2) and Y. N. Moschovakis (a3)

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