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Σ 1 definitions with parameters

Published online by Cambridge University Press:  12 March 2014

T. A. Slaman
T. A. Slaman*
Affiliation:
Department of Mathematics, University of Chicago, Chicago, Illinois 60637
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Abstract

Let p be a set. A function Φ is uniformly Σ 1(p) in every admissible set if there is a Σ 1 formula ϕ in the parameter p so that ϕ defines Φ in every Σ 1-admissible set which includes p. A theorem of Van de Wiele states that if Φ is a total function from sets to sets then Φ is uniformly Σ 1 in every admissible set if and only if it is E-recursive. A function is ESp -recursive if it can be generated from the schemes for E-recursion together with a selection scheme over the transitive closure of p. The selection scheme is exactly what is needed to insure that the ESP -recursively enumerable predicates are closed under existential quantification over the transitive closure of p. Two theorems are established: a) If the transitive closure of p is countable then a total function on sets is ESp -recursive if and only if it is uniformly Σ 1(p) in every admissible set. b) For any p, if Φ is a function on the ordinal numbers then Φ is ESP -recursive if and only if it is uniformly Σ 1(p) in every admissible set.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 51 , Issue 2 , June 1986 , pp. 453 - 461
Copyright
Copyright © Association for Symbolic Logic 1986

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Footnotes

1

The author was supported by NSF grant MCS-8404208.

References

REFERENCES

[Ga67] Gandy, R. O., Generalized recursive functionals of finite type and hierarchies of functions, Annales de la Faculté des Sciences de l'Université de Clermont-Ferrand, vol. 35, pp. 5–24.Google Scholar
[Ha73] Harrington, L. A., Contributions to recursion theory on higher types, Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, Massachusetts.Google Scholar
[Ho82] Hoole, M. R. R., Recursion on sets, Ph.D. thesis, Wolfson College, Oxford.Google Scholar
[Ke75] Kechris, A. S., The theory of countable analytic sets, Transactions of the American Mathematical Society, vol. 202, pp. 259–297.CrossRefGoogle Scholar
[No78] Normann, D., Set recursion, Generalized recursion theory. 2 (Fenstad, J. E. et al., editors), North-Holland, Amsterdam, pp. 303–320.Google Scholar
[Sa77] Sacks, G. E., The k-section of a type n object, American Journal of Mathematics, vol. 99, pp. 901–917.CrossRefGoogle Scholar
[Sa82] Sacks, G. E., Post's problem in E-recursion, Recursion theory, Proceedings of Symposia in Pure Mathematics, vol. 42, American Mathematical Society, Providence, Rhode Island, 1985, pp. 177–193.CrossRefGoogle Scholar
[Sl85] Slaman, T. A., Reflection and forcing in E-recursion theory, Annals of Pure and Applied Logic, vol. 28, pp. 1–28.Google Scholar
[Va81] van de Wiele, J., Dilateurs récursifs et récursivités généralisées, Thèse du troisième cycle, Université Paris-VIII, Paris.Google Scholar