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Every n-generic degree is a minimal cover of an n-generic degree

Published online by Cambridge University Press:  12 March 2014

Masahiro Kumabe
Masahiro Kumabe*
Affiliation:
Department of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455, E-mail: kumabe@math.umn.edu
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Extract

The notions of forcing and generic set were introduced by Cohen in 1963 to prove the independence of the Axiom of Choice and the Continuum Hypothesis in set theory. Let ω be the set of natural numbers, i.e., {0,1,2,3,…}. A string is a mapping from an initial segment of ω into {0,1}. We identify a set A ⊆ ω to with its characteristic function.

We now consider a set generic over the arithmetic sets. A set A ⊆ ω is called n-generic if it is Cohen-generic for n-quantifier arithmetic. This is equivalent to saying that for every -set of strings S, there is a σ ⊂ A such that σ ∈ S or (∀v ≥ σ)(v ∉ S). By degree we mean Turing degree (of unsolvability). We call a degree n-generic if it has an n-generic representative. For a degree a, let D(≤a) denote the set of degrees which are recursive in a.

Before Cohen's work, there was a precursor of the notion of forcing in recursion theory. Friedberg showed that for every degree b above the complete degree 0', i.e., the degree of a complete r.e. set, there is a degree a such that a′ = a ⋃ 0′ = b. He actually proved this result by using the notion of forcing for statements.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 58 , Issue 1 , March 1993 , pp. 219 - 231
Copyright
Copyright © Association for Symbolic Logic 1993

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References

REFERENCES

[1]Feferman, S., Some applications of the notion of forcing and generic sets. Fundamenta Mathematicae, vol. 55 (1965), pp. 325–345.Google Scholar
[2]Haught, C., The degrees below a I-generic degree < 0′, this Journal, vol. 51 (1986), 770–777.Google Scholar
[3]Jockusch, C. G., Degrees of generic sets, Recursion theory, its generalizations and applications, London Mathematical Society Lecture Notes, Cambridge University Press, Cambridge, 1980, pp. 110–139.CrossRefGoogle Scholar
[4]Jockusch, C. G. and Shore, R. A., REA operators, r.e. degrees and minimal covers, Recursion Theory (Nevode, A. and Shore, R. A., editors), Proceeding of Symposia in Pure Mathematics, vol. 42, American Mathematical Society, Providence, Rhode Island, 1985, pp. 3–11.CrossRefGoogle Scholar
[5]Jockusch, C. G. and Soare, R. I., Minimal covers and arithmetical sets, Proceedings of the American Mathematical Society, vol. 25 (1970), pp. 856–859.CrossRefGoogle Scholar
[6]Kumabe, M., Generic degrees are complemented, preprint.Google Scholar
[7]Kumabe, M., On decidability in the generic degrees, in preparation.Google Scholar
[8]Lerman, M., Degrees of unsolvability. Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1983.CrossRefGoogle Scholar
[9]Sacks, G. E., The recursively enumerable degrees are dense, Annals of Mathematics, vol. 80 (1964), pp. 300–312.CrossRefGoogle Scholar
[10]Shoenfield, J. R., A theorem on minimal degrees, this Journal, vol. 31 (1966), pp. 539–544.Google Scholar
[11]Soare, R. I., Recursively enumerable sets and degrees, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1987.CrossRefGoogle Scholar
[12]Spector, C., On degrees of recursive unsolvability, Annals of Mathematics, vol. 64 (1956), pp. 581–592.CrossRefGoogle Scholar
[13]Yates, C. E., Initial segment of degrees of unsolvability, Part II, minimal degrees, this Journal, vol. 35 (1970), pp. 243–266.Google Scholar
[14]Yates, C. E., Banach-Mazur games, comeager sets, and degrees of unsolvability. Mathematical Proceedings of the Cambridge Philosophical Society, vol. 79 (1976), pp. 195–220.CrossRefGoogle Scholar