Abstract
Using results from the local structure of the enumeration degrees we show the existence of prime ideals of enumeration degrees. We begin by showing that there exists a 1-generic enumeration degree which is noncuppable—and so properly downwards \(\Sigma^0_2\)—and low2. The notion of enumeration 1-genericity appropriate to positive reducibilities is introduced and a set A is defined to be symmetric enumeration 1-generic if both A and \(\ensuremath{\overline{A}} \) are enumeration 1-generic. We show that, if a set is 1-generic then it is symmetric enumeration 1-generic, and we prove that for any enumeration 1-generic set B the class \(\{\, X \,\mid \, \;\ensuremath{\negmedspace\leq_{\ensuremath{\mathrm{e}} }\negmedspace}\; B \,\}\) is uniform . Thus, picking 1-generic (from above) and defining it follows that every only contains sets. Since is properly \(\Sigma^0_2\) we deduce that contains no \(\Delta^0_2\) sets and so is itself properly .
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
Copestake, C.S.: 1-generic enumeration degrees below \({{\rm \bf 0}\sb e'}\). Mathematical logic. In: Proc. Summer Sch. Conf. Ded. 90th Anniv. Arend Heyting, Chaika/Bulg, pp. 257–265 (1988, 1990)
Cooper, S.B., McEvoy, K.: On minimal pairs of enumeration degrees. Journal of Symbolic Logic 50(4), 983–1001 (1985)
Cooper, S.B., Sorbi, A., Yi, X.: Cupping and noncupping in the enumeration degrees of \(\sigma^0_2\) sets. Annals of Pure and Applied Logic 82, 317–342 (1996)
Cooper, S.B., Li, A., Sorbi, A., Yang, Y.: Bounding and nonbounding minimal pairs in the enumeration degrees. Journal of Symbolic Logic 70(3), 741–766 (2005)
Friedberg, R.M., Rogers, H.: Reducibilities and completeness for sets of integers. Zeit. Math. Log. Grund. Math. 5, 117–125 (1959)
Griffith, E.J.: Limit lemmas and jump inversion in the enumeration degrees. Archive for Mathematical Logic 42, 553–562 (2003)
Harris, C.M.: Goodness in the enumeration and singleton degrees. Archive for Mathematical Logic 49(6), 673–691 (2010)
Harris, C.M.: Noncuppable enumeration degrees via finite injury. Journal of Logic and Computation (2011), doi:10.1093/logcom/exq044
Jockusch, C.G.: Semirecursive sets and positive reducibility. Trans. Amer. Math. Soc. 131, 420–436 (1968)
Lachlan, H., Shore, R.A.: The n-rea enumeration degrees are dense. Archive for Mathematical Logic 31, 277–285 (1992)
Soskova, M.I.: Genericity and Nonbounding. Journal of Logic and Computation 17, 1235–1255 (2007)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Badillo, L., Harris, C.M. (2012). An Application of 1-Genericity in the \(\Pi^0_2\) Enumeration Degrees. In: Agrawal, M., Cooper, S.B., Li, A. (eds) Theory and Applications of Models of Computation. TAMC 2012. Lecture Notes in Computer Science, vol 7287. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29952-0_56
Download citation
DOI: https://doi.org/10.1007/978-3-642-29952-0_56
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-29951-3
Online ISBN: 978-3-642-29952-0
eBook Packages: Computer ScienceComputer Science (R0)