Enumerations in computable structure theory

Annals of Pure and Applied Logic 136 (3):219-246 (2005)
  Copy   BIBTEX

Abstract

We exploit properties of certain directed graphs, obtained from the families of sets with special effective enumeration properties, to generalize several results in computable model theory to higher levels of the hyperarithmetical hierarchy. Families of sets with such enumeration features were previously built by Selivanov, Goncharov, and Wehner. For a computable successor ordinal α, we transform a countable directed graph into a structure such that has a isomorphic copy if and only if has a computable isomorphic copy.A computable structure is categorical if for all computable isomorphic copies of , there is an isomorphism from onto , which is . We prove that for every computable successor ordinal α, there is a computable, categorical structure, which is not relatively categorical. This generalizes the result of Goncharov that there is a computable, computably categorical structure, which is not relatively computably categorical.An additional relation R on the domain of a computable structure is intrinsically on if in all computable isomorphic copies of , the image of R is . We prove that for every computable successor ordinal α, there is an intrinsically relation on a computable structure, which not relatively intrinsically . This generalizes the result of Manasse that there is an intrinsically computably enumerable relation on a computable structure, which is not relatively intrinsically computably enumerable.The dimension of a structure is the number of computable isomorphic copies, up to isomorphisms. We prove that for every computable successor ordinal α and every n≥1, there is a computable structure with dimension n. This generalizes the result of Goncharov that there is a structure of computable dimension n for every n≥1.Finally, we prove that for every computable successor ordinal α, there is a countable structure with isomorphic copies in just the Turing degrees of sets X such that relative to X is not . In particular, for every finite n, there is a structure with isomorphic copies in exactly the non- Turing degrees. This generalizes the result obtained by Wehner, and independently by Slaman, that there is a structure with isomorphic copies in exactly the nonzero Turing degrees

Links

PhilArchive



    Upload a copy of this work     Papers currently archived: 92,931

External links

Setup an account with your affiliations in order to access resources via your University's proxy server

Through your library

Similar books and articles

Prime models of finite computable dimension.Pavel Semukhin - 2009 - Journal of Symbolic Logic 74 (1):336-348.
Spectra of Structures and Relations.Valentina S. Harizanov & Russel G. Miller - 2007 - Journal of Symbolic Logic 72 (1):324 - 348.

Analytics

Added to PP
2013-10-30

Downloads
68 (#245,269)

6 months
30 (#108,804)

Historical graph of downloads
How can I increase my downloads?

Author's Profile

Valentina Harizanov
George Washington University

References found in this work

Computable structures and the hyperarithmetical hierarchy.C. J. Ash - 2000 - New York: Elsevier. Edited by J. Knight.
Generic copies of countable structures.Chris Ash, Julia Knight, Mark Manasse & Theodore Slaman - 1989 - Annals of Pure and Applied Logic 42 (3):195-205.
[Omnibus Review].H. Jerome Keisler - 1970 - Journal of Symbolic Logic 35 (2):342-344.
Pairs of recursive structures.C. J. Ash & J. F. Knight - 1990 - Annals of Pure and Applied Logic 46 (3):211-234.

View all 19 references / Add more references