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α-degrees of α-theories1

Published online by Cambridge University Press:  12 March 2014

George Metakides
Affiliation:
Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

Extract

Let α be a limit ordinal with the property that any “recursive” function whose domain is a proper initial segment of α has its range bounded by α. α is then called admissible (in a sense to be made precise later) and a recursion theory can be developed on it (α-recursion theory) by providing the generalized notions of α-recursively enumerable, α-recursive and α-finite. Takeuti [12] was the first to study recursive functions of ordinals, the subject owing its further development to Kripke [7], Platek [8], Kreisel [6], and Sacks [9].

Infinitary logic on the other hand (i.e., the study of languages which allow expressions of infinite length) was quite extensively studied by Scott [11], Tarski, Kreisel, Karp [5] and others. Kreisel suggested in the late '50's that these languages (even which allows countable expressions but only finite quantification) were too large and that one should only allow expressions which are, in some generalized sense, finite. This made the application of generalized recursion theory to the logic of infinitary languages appear natural. In 1967 Barwise [1] was the first to present a complete formalization of the restriction of to an admissible fragment (A a countable admissible set) and to prove that completeness and compactness hold for it. [2] is an excellent reference for a detailed exposition of admissible languages.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1972

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Footnotes

1

This research was done as part of the author's Doctoral Dissertation (Cornell University, 1971) which was partially supported by NSF grant GP-22719 and was directed by Professor Anil Nerode, to whom, along with Professors R. Platek and M. D. Morley, the author expresses his thanks for numerous conversations. The author also wishes to thank the referee of this paper for his valuable suggestions.

References

[1]Barwise, Jon, Infinitary logic and admissible sets, Doctoral Dissertation, Stanford University, Stanford, Calif., 1967.Google Scholar
[2]Barwise, Jon, Infinitary logic and admissible sets, this Journal, vol. 34 (1969), pp. 226252.Google Scholar
[3]Erdös, P. and Hajnal, A., On a classification of denumerable order types and an application to the partition calculus, Fundamenta Mathematicae, vol. 51 (1962), pp. 116129.CrossRefGoogle Scholar
[4]Feferman, S., Degrees of unsolvability associated with classes of formalized theories, this Journal, vol. 22 (1957), pp. 161175.Google Scholar
[5]Karp, C., Languages with expressions of infinite length, North-Holland, Amsterdam, 1964.Google Scholar
[6]Kreisel, G. and Sacks, G. E., Metarecursive sets, this Journal, vol. 30 (1965), pp. 318338.Google Scholar
[7]Kripke, S., Transfinite recursions on admissible ordinals, this Journal, vol. 29 (1964), pp. 161162. Abstracts.Google Scholar
[8]Platek, R., Foundations of recursion theory, Ph.D. Thesis, Stanford University, Stanford, Calif., 1966.Google Scholar
[9]Sacks, G. E., Metarecursion theory, Sets, models and recursion theory, North-Holland, Amsterdam, 1967, pp. 243263.CrossRefGoogle Scholar
[10]Sacks, G. E., Higher recursion theory, Springer-Verlag (to appear).Google Scholar
[11]Scott, D., Logic with denumerably long formulas and finite strings of quantifiers, The theory of models (Addison, J., Henkin, L. and Tarski, A., Editors), North-Holland, Amsterdam, 1965, pp. 329341.Google Scholar
[12]Takeuti, G., On the recursive functions of ordinal numbers, Journal of the Mathematical Society of Japan, vol. 12 (1960), pp. 119128.CrossRefGoogle Scholar
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