Journal of Symbolic Logic

Up to equimorphism, hyperarithmetic is recursive

Antonio Montalbán

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Two linear orderings are equimorphic if each can be embedded into the other. We prove that every hyperarithmetic linear ordering is equimorphic to a recursive one.

On the way to our main result we prove that a linear ordering has Hausdorff rank less than ω₁CK if and only if it is equimorphic to a recursive one. As a corollary of our proof we prove that, given a recursive ordinal α, the partial ordering of equimorphism types of linear orderings of Hausdorff rank at most α ordered by embeddablity is recursively presentable.

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J. Symbolic Logic, Volume 70, Issue 2 (2005), 360-378.

First available in Project Euclid: 1 July 2005

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Montalbán, Antonio. Up to equimorphism, hyperarithmetic is recursive. J. Symbolic Logic 70 (2005), no. 2, 360--378. doi:10.2178/jsl/1120224717.

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  • Uri Abraham and Robert Bonnet Hausdorff's theorem for posets that satisfy the finite antichain property, Fundamenta Mathematicae, vol. 159 (1999), no. 1, pp. 51--69.
  • C. J. Ash and J. Knight Computable structures and the hyperarithmetical hierarchy, Elsevier Science,2000.
  • R. Bonnet and M. Pouzet Linear extensions of ordered sets, Ordered sets (Banff, Alta., 1981), NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, vol. 83, Reidel, Dordrecht,1982, pp. 125--170.
  • P. Clote The metamathematics of scattered linear orderings, Archive for Mathematical Logic, vol. 29 (1989), no. 1, pp. 9--20.
  • R. G. Downey Computability theory and linear orderings, Handbook of recursive mathematics, vol. 2 (Yu. L. Ershov et al., editors), Studies in Logic and the Foundations of Mathematics, vol. 139, North-Holland, Amsterdam,1998, pp. 823--976.
  • Rodney G. Downey, Denis R. Hirschfeldt, Steffen Lempp, and Reed Solomon Computability-theoretic and proof-theoretic aspects of partial and linear orderings, Israel Journal of Mathematics, vol. 138 (2003), pp. 271--352.
  • Lawrence Feiner Orderings and Boolean Algebras not isomorphic to recursive ones, Ph.D. thesis, MIT, Cambridge, MA,1967.
  • Roland Fraï"ssé Sur la comparaison des types d'ordres, Comptes Rendus de l'Académie des Sciences. Paris, vol. 226 (1948), pp. 1330--1331.
  • G. Hessenberg Grundbegriffe der Mengenlehre, Abhandlungen der Fries'schen Schule N.S., vol. 1 (1906), pp. 479--706.
  • Carl G. Jockusch, Jr. and Robert I. Soare Degrees of orderings not isomorphic to recursive linear orderings, Annals of Pure and Applied Logic, vol. 52 (1991), no. 1--2, pp. 39--64, International Symposium on Mathematical Logic and its Applications (Nagoya, 1988).
  • Pierre Jullien Contribution à l'étude des types d'ordre dispersés, Ph.D. thesis, Marseille,1969.
  • Richard Laver On Fraïssé's order type conjecture, Annals of Mathematics. Second Series, vol. 93 (1971), pp. 89--111.
  • Manuel Lerman On recursive linear orderings, Logic Year 1979--80 (Proceedings of Seminars and Conferences in Mathematical Logic, University of Connecticut, Storrs, Connecticut, 1979/80), Lecture Notes in Mathematics, vol. 859, Springer, Berlin,1981, pp. 132--142.
  • Alberto Marcone WQO and BQO theory in subsystems of second order arithmetic, Reverse mathematics (S. Simpson, editor), Lecture Notes in Logic, vol. 21, AK Peters,2005, pp. 303--330.
  • E. C. Milner Basic wqo- and bqo-theory, Graphs and order (Banff, Alta., 1984), NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, vol. 147, Reidel, Dordrecht,1985, pp. 487--502.
  • Antonio Montalbán Equivalence between Fraïssé's conjecture and Jullien's theorem, To appear.
  • C. St. J. A. Nash-Williams On better-quasi-ordering transfinite sequences, Proceedings of the Cambridge Philosophical Society, vol. 64 (1968), pp. 273--290.
  • Joseph Rosenstein Linear orderings, Academic Press, New York -- London,1982.
  • Gerald E. Sacks Higher recursion theory, Perspectives in Mathematical Logic, Springer-Verlag, Berlin,1990.
  • Richard A. Shore On the strength of Fraïssé's conjecture, Logical methods (Ithaca, NY, 1992), Progress in Computer Science and Applied Logic, vol. 12, Birkhäuser Boston, Boston, MA,1993, pp. 782--813.
  • Stephen G. Simpson Subsystems of second order arithmetic, Springer,1999.
  • Clifford Spector Recursive well-orderings, Journal of Symbolic Logic, vol. 20 (1955), pp. 151--163.