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The Meta-R.E. sets, but not the Π11 sets, can be enumerated without repetition

Published online by Cambridge University Press:  12 March 2014

James C. Owings Jr.
James C. Owings Jr.*
Affiliation:
University of Maryland
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Extract

In this paper we investigate the possibility of extending Friedberg's enumeration of the recursively enumerable (r.e.) sets without duplication [1, p. 312] to meta-recursion theory. It turns out that all of our proposed extensions are impossible save one: the metarecursively enumerable (meta-r.e.) sets can be enumerated without duplication, but only if all the recursive ordinals are used as indices (Theorems 1 and 2). The sets cannot be so enumerated, even if the index set is all recursive ordinals (Theorems 3 and 4). As a corollary, one proves there is no predicate P(n, x) with the property that for each set A there is exactly one integer n for which A = {x ∣ P(n, x)}. We also discuss enumerations of nonempty, infinite, and coinfinite and meta-r.e. sets.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 35 , Issue 2 , June 1970 , pp. 223 - 229
Copyright
Copyright © Association for Symbolic Logic 1970

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References

[1]Friedberg, R. M., Three theorems on recursive enumeration, this Journal, vol. 23 (1958), pp. 309–316.Google Scholar
[2]Kreisel, G. and Sacks, G. E., Metarecursive sets, this Journal, vol. 30 (1965), pp. 318–338.Google Scholar
[3]Owings, J. C. Jr., sets, ω-sets, and metacompleteness, this Journal, vol. 34 (1969), pp. 194–204.Google Scholar