Abstract
We first prove a theorem about reals (subsets of \(\mathbb {N}\)) and classes of reals: If a real X is \(\Sigma _{1}^{1}\) in every member G of a nonempty \(\Sigma _{1}^{1}\) class \(\mathcal {K}\) of reals then X is itself \(\Sigma _{1}^{1}\). We also explore the relationship between this theorem, various basis results in hyperarithmetic theory and omitting types theorems in \(\omega \)-logic. We then prove the analog of our first theorem for classes of reals: If a class \(\mathcal {A}\) of reals is \(\Sigma _{1}^{1}\) in every member of a nonempty \(\Sigma _{1}^{1}\) class \(\mathcal {B}\) of reals then \(\mathcal {A}\) is itself \(\Sigma _{1}^{1}\).
R.A. Shore—Partially supported by NSF Grant DMS-1161175. The last two authors began their work on this paper at a workshop of the Institute for Mathematical Sciences of the National University of Singapore which also partially supported them.
T.A. Slaman—Partially supported by NSF Grant DMS-1301659 and by the Institute for Mathematical Sciences of the National University of Singapore.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Andrews, U., Miller, J.S.: Spectra of theories and structures. Proc. Am. Math. Soc. 143, 1283–1298 (2015)
Hinman, P.G.: Recursion Theoretic Hierarchies. Perspectives in Mathematical Logic. Springer, Berlin (1978)
Miller, A.W.: Descriptive Set Theory and Forcing. Lecture Notes in Logic. Springer, Berlin (1995)
Moschovakis, Y.N.: Descriptive Set Theory. Studies in Logic and the Foundations of Mathematics, vol. 100. North-Holland, Amsterdam (1980)
Sacks, G.E.: Higher Recursion Theory. Perspectives in Mathematical Logic. Springer, Berlin (1990)
Simpson, S.G.: Subsystems of Second Order Arithmetic. Perspectives in Logic, 2nd edn. ASL and Cambridge University Press, New York (2009)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this chapter
Cite this chapter
Harrington, L., Shore, R.A., Slaman, T.A. (2017). \(\Sigma _{1}^{1}\) in Every Real in a \(\Sigma _{1}^{1}\) Class of Reals Is \(\Sigma _{1}^{1}\) . In: Day, A., Fellows, M., Greenberg, N., Khoussainov, B., Melnikov, A., Rosamond, F. (eds) Computability and Complexity. Lecture Notes in Computer Science(), vol 10010. Springer, Cham. https://doi.org/10.1007/978-3-319-50062-1_26
Download citation
DOI: https://doi.org/10.1007/978-3-319-50062-1_26
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-50061-4
Online ISBN: 978-3-319-50062-1
eBook Packages: Computer ScienceComputer Science (R0)