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The least α for which E(α) is inadmissible

Part of: Computability and recursion theory

Published online by Cambridge University Press:  09 April 2009

M. R. R. Hoole
M. R. R. Hoole
Affiliation:
Department of Mathematics, University of Jaffna, Jaffna, Sri Lanka
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Abstract

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This paper attempts to classify the least ordinal α0 for which E(α0) (the E closure of α0 ∪ {α0}) is inadmissible. Among the result proved are (i)Lα0 = ZFC-; (ii)α0 is very large in comparison with the least ordinal satifying (i); (iii) (α0, α] marks precisely an ω-Gap, where α¯ = E(α0) ∩ ON; (iv) the Kr-sequence of α0 has length ω.

MSC classification

Secondary: 03D25: Recursively (computably) enumerable sets and degrees
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 41 , Issue 2 , October 1986 , pp. 143 - 151
Copyright
Copyright © Australian Mathematical Society 1986

References

[1]Barwise, J., Gandy, R. and Moschovakis, Y., “The next admissible set”, J. Symbolic Logic 36 (1971), 108–120.Google Scholar
[2]Fenstad, J. E., “On axiomatizing recursion theory”, pp. 385–404, in Generalized Recursion Theory, Proceedings of the 1972 Oslo Symposium, eds. Fenstad, J. E. and Hinman, P. G., North-Holland, 1974.Google Scholar
[3]Normann, D., “Set recursion”, pp. 303–320, in Generalized Recursion Theory II, Proceedings of the 1977 Oslo Symposium, eds. Fenstad, J. E., Gandy, R. O. and Sacks, G. E., North-Holland, 1978.Google Scholar
[4]Gostanian, R., “The next admissible ordinal”, Ann. Math. Logic 17 (1979), 171–203.CrossRefGoogle Scholar
[5]Marek, W. and Srebrny, M., “Gaps in the constructible universe”, Ann. Math. Logic 6 (1974), 359–394.CrossRefGoogle Scholar
[6]Moldestad, J., Computations in higher types (Lecture Notes in Mathematics 574, Springer-Verlag, Berlin, 1977).Google Scholar
[7]Slaman, T., Contributions to E-recursion theory, Ph.D. Thesis, MIT, 1981.Google Scholar
[8]Jensen, R. B., “The fine structure of the constructible hierarchy”, Ann. Math. Logic 4 (1972), 229–308.CrossRefGoogle Scholar
[9]Sacks, G. E., “The K-section of a type n object”, Amer. J. Math. 99 (1977), 901–917.CrossRefGoogle Scholar
[10]Grilliot, T., “Inductive definitions and computability”, Trans. Amer. Math. Soc. 158 (1971), 309–317.CrossRefGoogle Scholar