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Up to equimorphism, hyperarithmetic is recursive

  • Antonio Montalbán (a1)

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 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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[Clo89] P. Cloth , The metamathematics of scattered linear orderings, Archive for Mathematical Logic, vol. 29 (1989), no. 1, pp. 920.

[Clo90] P. Cloth , The metamathematics of Fraïssé's order type conjecture, Recursion theory week (Oberwolfach, 1989), Lecture Notes in Mathematics, vol. 1432, Springer, Berlin, 1990, pp. 4156.

[DHLS03] Rodney G. Downey , Denis R. Hirschfeldt , Steffen Lempp , and Reed Solomon , Computahility-theoretic and proof-theoretic aspects of partial and linear orderings, Israel Journal of Mathematics, vol. 138 (2003), pp. 271352.

[JS91] 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. 3964, International Symposium on Mathematical Logic and its Applications (Nagoya, 1988).

[Lav71] Richard Laver , On Fruïssé's order type conjecture, Annals of Mathematics. Second Series, vol. 93 (1971), pp. 89111.

[Sac90] Gerald E. Sacks , Higher recursion theory, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1990.

[Sim99] Stephen G. Simpson , Subsystems of second order arithmetic, Springer, 1999.

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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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