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The Π3-theory of the -enumeration degrees is undecidable

Published online by Cambridge University Press:  12 March 2014

Thomas F. Kent
Thomas F. Kent*
Affiliation:
Department of Mathematics, Brigham Young University, Provo, UT 84602., USA, E-mail: kent@math.byu.edu
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Abstract

We show that in the language of { ≤ }. the Π3-fragment of the first order theory of the -enumeration degrees is undecidable. We then extend this result to show that the Π3-theory of any substructure of the enumeration degrees which contains the -degrees is undecidable.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 71 , Issue 4 , December 2006 , pp. 1284 - 1302
Copyright
Copyright © Association for Symbolic Logic 2006

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References

REFERENCES

[1]Ahmad, S., Some results on enumeration reducibility, Ph.D. thesis, Simon Frasier University, 1989.Google Scholar
[2]Ahmad, S., Embedding the diamond in the enumeration degrees, this Journal, vol. 56 (1991), no. 1, pp. 195–212.Google Scholar
[3]Ahmad, S. and Lachlan, A. H., Some special pairs of e-degrees, Mathematical Logic Quarterly, vol. 44 (1998), no. 4, pp. 431–449.CrossRefGoogle Scholar
[4]Case, J., Enumeration reducibility and partial degrees, Archive for Mathematical Logic, vol. 2 (1971), pp. 419–439.CrossRefGoogle Scholar
[5]Cooper, S., Partial degrees and the density problem, part II: The enumeration degrees of the sets are dense, this Journal, vol. 49 (1984), no. 2, pp. 503–513.Google Scholar
[6]Lachlan, A. H., Lower bounds for pairs of recursively enumerable degrees, Proceedings of the London Mathematical Society, vol. 16 (1966), pp. 537–569.CrossRefGoogle Scholar
[7]Lachlan, A. H. and Shore, R., The n-rea enumeration degrees are dense, Archive for Mathematical Logic, vol. 31 (1992), no. 4, pp. 277–285.CrossRefGoogle Scholar
[8]Lempp, S. and Nies, A., The undecidability of the Π4-theory for the r.e. wtt- and Turing-degrees, this Journal, vol. 60 (1995), no. 4, pp. 1118–1136.Google Scholar
[9]Lempp, S., Nies, A., and Slaman, T., The Π3-theory of the computably enumerable Turing degrees is undecidable, Transactions of the American Mathematical Society, vol. 350 (1998), no. 7, pp. 2719–2736.CrossRefGoogle Scholar
[10]Lempp, S., Slaman, T., and Sorbi, S., On extensions of embeddings into the enumeration degrees of the -sets, Journal of Mathematical Logic, vol. 5 (2005), no. 2, pp. 247–298.CrossRefGoogle Scholar
[11]Lempp, S. and Sorbi, A., Embedding finite lattices into the enumeration degrees, this Journal, vol. 67 (2002), no. 1. pp. 69–90.Google Scholar
[12]McEvoy, K., Jumps of quasi-minimal enumeration degrees, this Journal, vol. 50 (1985), no. 3, pp. 839–848.Google Scholar
[13]McEvoy, K. and Cooper, S., On minimal pairs of enumeration-degrees, this Journal, vol. 50 (1985), no. 4, pp. 839–848.Google Scholar
[14]Nies, A., Undecidable fragments of elementary theories, Algebra Universalis, vol. 35 (1996), no. 1, pp. 8–33.CrossRefGoogle Scholar
[15]Sacks, G. E., On the degrees less than 0′, Annals of Mathematics, vol. 77 (1963), pp. 211–231.CrossRefGoogle Scholar
[16]Slaman, T. and Woodin, W., Definability in the enumeration degrees, Archive for Mathematical Logic, vol. 36 (1997), no. 4-5, pp. 255–267, Sacks Symposium (Cambridge, MA, 1993).CrossRefGoogle Scholar
[17]Soare, R. I., Recursively enumerable sets and degrees, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1987.CrossRefGoogle Scholar