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Π11 relations and paths through

Published online by Cambridge University Press:  12 March 2014

Sergey S. Goncharov ,
Valentina S. Harizanov ,
Julia F. Knight  and
Richard A. Shore
Sergey S. Goncharov
Affiliation:
Academy of Sciences, Siberian Branch, Mathematical Institute, 630090 Novosibirsk, Russia, E-mail: gonchar@math.nsc.ru
Valentina S. Harizanov
Affiliation:
Department of Mathematics, The George Washington University, Washington, DC. 20052, U.S.A., E-mail: harizanv@gwu.edu
Julia F. Knight
Affiliation:
Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, U.S.A., E-mail: julia.f.knight.1@nd.edu
Richard A. Shore
Affiliation:
Department of Mathematics, Cornell University, Ithaca, NY 14853, U.S.A., E-mail: shore@math.cornell.edu
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Extract

When bounds on complexity of some aspect of a structure are preserved under isomorphism, we refer to them as intrinsic. Here, building on work of Soskov [34], [33], we give syntactical conditions necessary and sufficient for a relation to be intrinsically on a structure. We consider some examples of computable structures and intrinsically relations R. We also consider a general family of examples of intrinsically relations arising in computable structures of maximum Scott rank.

For three of the examples, the maximal well-ordered initial segment in a Harrison ordering, the superatomic part of a Harrison Boolean algebra, and the height-possessing part of a Harrison p-group, we show that the Turing degrees of images of the relation in computable copies of the structure are the same as the Turing degrees of paths through Kleene's . With this as motivation, we investigate the possible degrees of these paths. We show that there is a path in which ∅′ is not computable. In fact, there is one in which no noncomputable hyperarithmetical set is computable. There are paths that are Turing incomparable, or Turing incomparable over a given hyperarithmetical set. There is a pair of paths whose degrees form a minimal pair. However, there is no path of minimal degree.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 69 , Issue 2 , June 2004 , pp. 585 - 611
Copyright
Copyright © Association for Symbolic Logic 2004

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