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The computable dimension of trees of infinite height

Published online by Cambridge University Press:  12 March 2014

Russell Miller
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
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Abstract

We prove that no computable tree of infinite height is computably categorical, and indeed that all such trees have computable dimension ω. Moreover, this dimension is effectively ω, in the sense that given any effective listing of computable presentations of the same tree, we can effectively find another computable presentation of it which is not computably isomorphic to any of the presentations on the list.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 70 , Issue 1 , March 2005 , pp. 111 - 141
Copyright
Copyright © Association for Symbolic Logic 2005

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References

REFERENCES

[1]Ash, C. J., Categoricity in hyperarithmetical degrees, Annals of Pure and Applied Logic, vol. 34 (1987), pp. 1–14.CrossRefGoogle Scholar
[2]Crossley, J. N., Manaster, A. B., and Moses, M. F., Recursive categoricity and recursive stability, Annals of Pure and Applied Logic, vol. 31 (1986), pp. 191–204.CrossRefGoogle Scholar
[3]Downey, R., On presentations of algebraic structures, Complexity, logic, and recursion theory (Sorbi, A., editor), Dekker, New York, 1997, pp. 157–205.Google Scholar
[4]Ershov, Y. L. and Goncharov, S. S., Constructive models, Kluwer Academic/Plenum Press, New York, 2000.CrossRefGoogle Scholar
[5]Gallier, J. H., What's so special about Kruskal's theorem and the ordinal Γ0?, Annals of Pure and Applied Logic, vol. 53 (1991), pp. 199–260.CrossRefGoogle Scholar
[6]Goncharov, S. S., Autostability and computable families of constructivizations, Algebra and Logic, vol. 14 (1975), pp. 647–680 (Russian), pp. 392–409 (English translation).CrossRefGoogle Scholar
[7]Goncharov, S. S., The problem of the number of non-self-equivalent constructivizations, Algebra i Logika, vol. 19 (1980).Google Scholar
[8]Goncharov, S. S., Nonequivalent constructivizations, Proceedings of the Mathematical Institute of Siberia Branch of Academic Sciences, Nauka, Novosibirsk, 1982.Google Scholar
[9]Goncharov, S. S., Autostable models and algorithmic dimensions, Handbook of recursive mathematics (Ershov, Yu. L. et al., editors), vol. 1, Elsevier, Amsterdam, 1998, pp. 261–287.Google Scholar
[10]Goncharov, S. S. and Dzgoev, V. D., Autostability of models, Algebra and Logic, vol. 19 (1980), pp. 45–58 (Russian), pp. 28–37 (English translation).Google Scholar
[11]Khoussainov, B. and Shore, R. A., Computable isomorphisms, degree spectra of relations, and Scott families, Annals of Pure and Applied Logic, vol. 93 (1998), pp. 153–193.CrossRefGoogle Scholar
[12]Khoussainov, B., Effective model theory: The number of models and their complexity, Models and computability: Invited papers from Logic Colloquium '97 (Cooper, S. B. and Truss, J. K., editors), London Mathematical Society Lecture Notes Series, vol. 259, Cambridge University Press, Cambridge, 1999, pp. 193–240.CrossRefGoogle Scholar
[13]Kruskal, J. B., Well quasi-ordering, the tree theorem, and Vázsonyi's conjecture, Transactions of the American Mathematical Society, vol. 95 (1960), pp. 210–225.Google Scholar
[14]Lempp, S., McCoy, C., Miller, R. G., and Solomon, R., Computable categoricity of trees offinite height, this Journal, vol. 70 (2005), pp. 151–215.Google Scholar
[15]Nash-Williams, C. St. J. A., On well-quasi-ordering finite trees, Proceedings of the Cambridge Philosophical Society, vol. 59 (1963), pp. 833–835.CrossRefGoogle Scholar
[16]Remmel, J. B., Recursive isomorphism types of recursive Boolean algebras, this Journal, vol. 46 (1981), pp. 572–594.Google Scholar
[17]Remmel, J. B., Recursively categorical linear orderings, Proceedings of the American Mathematical Society, vol. 83 (1981), pp. 387–391.CrossRefGoogle Scholar
[18]Simpson, S. G., Nonprovability of certain combinatorial properties of finite trees, Harvey Friedman's research on the foundations of mathematics (Harrington, L. A., Morley, M. D., Scedrov, A., and Simpson, S. G., editors), North-Holland, Amsterdam, 1985, pp. 87–117.CrossRefGoogle Scholar
[19]Soare, R. I.. Recursively enumerable sets and degrees, Springer-Verlag, New York, 1987.CrossRefGoogle Scholar