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Computable categoricity of trees of finite height

Published online by Cambridge University Press:  12 March 2014

Steffen Lempp
Affiliation:
Department of Mathematics, University of Wisconsin-Madison, 480 Lincoln Drive, Madison, Wisconsin 53706-1388, USA, E-mail: lempp@math.wisc.edu
Charles McCoy
Affiliation:
Moreau Seminary, Notre Dame, Indiana 46556, USA, E-mail: cmccoyl@nd.edu
Russell Miller
Affiliation:
Department of Mathematics, Queens College —C.U.N.Y., 65-30 Kissena Blvd., Flushing, New York 11367, USA, E-mail: rmiller@forbin.qc.edu
Reed Solomon
Affiliation:
Department of Mathematics, University of Connecticut, 196 Auditorium Road, Storrs, Connecticut 06269-3009, USA, E-mail: solomon@math.uconn.edu

Abstract

We characterize the structure of computably categorical trees of finite height, and prove that our criterion is both necessary and sufficient. Intuitively, the characterization is easiest to express in terms of isomorphisms of (possibly infinite) trees, but in fact it is equivalent to a -condition. We show that all trees which are not computably categorical have computable dimension ω. Finally, we prove that for every n ≥ 1 in ω, there exists a computable tree of finite height which is Σ30-categorical but not Δn3-categorical

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2005

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