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COMPUTABILITY AND UNCOUNTABLE LINEAR ORDERS I: COMPUTABLE CATEGORICITY

  • NOAM GREENBERG (a1), ASHER M. KACH (a2), STEFFEN LEMPP (a3) and DANIEL D. TURETSKY (a4)

Abstract

We study the computable structure theory of linear orders of size $\aleph _1 $ within the framework of admissible computability theory. In particular, we characterize which of these linear orders are computably categorical.

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[1]Downey, Rodney G., Computability theory and linear orderings. In Handbook of recursive mathematics, Vol. 2, Studies in Logic and the Foundations of Mathematics, vol. 139 pages 823976. North-Holland, Amsterdam, 1998.
[2]Ben, Dushnik and Miller, E. W., Concerning similarity transformations of linearly ordered sets. Bulletin of American Mathematical Society, vol. 46 (1940), pp. 322326.
[3]Dzgoev, Valeri D., On constructivizations of some structures. Akad. Nauk SSSR, Sibirsk. Otdel., Novosibirsk, 1978. manuscript deposited at VINITI on July 26, 1978, Deposition No. 1606–79.
[4]Goncharov, Sergey S. and Dzgoev, Valeri D., Autostability of models. Algebra i Logika, vol. 19 (1980), no. 1, pp. 4558, 132.
[5]Greenberg, Noam, The role of true finiteness in the admissible recursively enumerable degrees. Memoirs of the American Mathematical Society, 181(854):vi+99, 2006.
[6]Greenberg, Noam, Kach, Asher M., Lempp, Steffen and Turetsky, Daniel D., Computability and uncountable linear orders, part II: degree spectra, this Journal, accepted.
[7]Greenberg, Noam and Knight, Julia F.. Computable structure theory using admissible recursion theory on ω 1, Effective mathematics of the uncountable (Greenberg, Hirschfeldt, Miller, Hamkins, editors), Lecture Notes in Logic, Association for Symbolic Logic and Cambridge University Press, pp. 5080, 2013, available at http://www.cambridge.org/nz/academic/subjects/mathematics/logic-categories-and-sets/effective-mathematics-uncountable.
[8]Kreisel, Georg, Set theoretic problems suggested by the notion of potential totality. Infinitistic Methods (Proc. Sympos. Foundations of Math., Warsaw, 1959), pp. 103140, Pergamon, Oxford, 1961.
[9]Kreisel, Georg and Sacks, Gerald E.. Metarecursive sets I, II (abstracts), this Journal, vol. 28 (1963), pp. 304–305.
[10]Kreisel, Georg, Metarecursive sets, this Journal, vol. 30 (1965), pp. 318–338.
[11]Kripke, Saul A., Transfinite recursion on admissible ordinals I, II (abstracts), this Journal, vol. 29 (1964), pp. 161–162.
[12]Lerman, Manuel, Maximal α-r.e. sets. Transactions of the American Mathematical Society, vol. 188 (1974), pp. 341386.
[13]Lerman, Manuel and Simpson, Stephen G., Maximal sets in α-recursion theory. Israel Journal of Mathematics, vol. 14 (1973), pp. 236247.
[14]Platek, Richard A., Foundations of recursion theory, Ph.D. thesis, Stanford Univeristy, Stanford, CA, 1965.
[15]Remmel, Jeffrey B., Recursively categorical linear orderings. Proceedings of the American Mathematical Society, vol. 83 (1981), no. 2, pp. 387391.
[16]Richter, Linda Jean, Degrees of structures, this Journal, vol. 46 (1981), no. 4, pp. 723–731.
[17]Rosenstein, Joseph G., Linear orderings, Pure and Applied Mathematics, vol. 98, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1982.
[18]Sacks, Gerald E., Higher recursion theory, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1990.
[19]Takeuti, Gaisi, On the recursive functions of ordinal numbers. Journal of the Mathematical Society of Japan, vol. 12 (1960), pp. 119128.
[20]Takeuti, Gaisi, A formalization of the theory of ordinal numbers, this Journal, vol. 30 (1965), pp. 295–317.
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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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