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Up to equimorphism, hyperarithmetic is recursive

Published online by Cambridge University Press:  12 March 2014

Antonio Montalbán
Antonio Montalbán*
Affiliation:
Department of Mathematics, Cornell University, Ithaca, NY 14853, USA, E-mail: antonio@math.cornell.edu URL: www.math.cornell.edu/~antonio
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Abstract

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.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 70 , Issue 2 , June 2005 , pp. 360 - 378
Copyright
Copyright © Association for Symbolic Logic 2005

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References

REFERENCES

[AB99] Abraham, Uri and Bonnet, Rorert, Hausdorff's theorem for posets that satisfy the finite antichain property, Fundamenta Mathematicae, vol. 159 (1999), no. 1, pp. 51–69.CrossRefGoogle Scholar
[AK00] Ash, C. J. and Knight, J., Computable structures and the hyperarithmetical hierarchy, Elsevier Science, 2000.Google Scholar
[BP82] Bonnet, R. and Pouzet, M., 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.CrossRefGoogle Scholar
[Clo89] Cloth, P., The metamathematics of scattered linear orderings, Archive for Mathematical Logic, vol. 29 (1989), no. 1, pp. 9–20.CrossRefGoogle Scholar
[Clo90] Cloth, P., The metamathematics of Fraïssé's order type conjecture, Recursion theory week (Oberwolfach, 1989), Lecture Notes in Mathematics, vol. 1432, Springer, Berlin, 1990, pp. 41–56.CrossRefGoogle Scholar
[Dow98] Downey, R. G., Computahility theory and linear orderings, Handbook of recursive mathematics, vol. 2 (Ershov, Yu. L. et al., editors). Studies in Logic and the Foundations of Mathematics, vol. 139, North-Holland, Amsterdam, 1998, pp. 823–976.Google Scholar
[DHLS03] Downey, Rodney G., Hirschfeldt, Denis R., Lempp, Steffen, and Solomon, Reed, Computahility-theoretic and proof-theoretic aspects of partial and linear orderings, Israel Journal of Mathematics, vol. 138 (2003), pp. 271–352.CrossRefGoogle Scholar
[Fei67] Feiner, Lawrence, Orderings and Boolean Algebras not isomorphic to recursive ones, Ph.D. thesis, MIT, Cambridge, MA, 1967.Google Scholar
[Fei70] Feiner, Lawrence, Hierarchies of Boolean algebras, this Journal, vol. 35 (1970), pp. 365–374.Google Scholar
[Fra48] Fraïssé, Roland, Sur la comparaison des types d'ordres, Comptes Rendus de l'Académie des Sciences. Paris vol. 226 (1948), pp. 1330–1331.Google Scholar
[Hes06] Hessenberg, G., Grundbegriffe der Mengenlehre, Abhandlungen der Fries'schen Schule N.S., vol. 1 (1906), pp. 479–706.Google Scholar
[JS91] Jockusch, Carl G. Jr., and Soare, Robert I., 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).CrossRefGoogle Scholar
[Jul69] Jlillien, Pierre, Contribution à l'étude des types d'ordre dispersés. Ph.D, thesis, Marseille, 1969.Google Scholar
[Lav71] Laver, Richard, On Fruïssé's order type conjecture, Annals of Mathematics. Second Series, vol. 93 (1971), pp. 89–111.CrossRefGoogle Scholar
[Ler81] Lerman, Manuel, On recursive linear orderings, Logic Year 1979–80 (Proceedings of Seminars and Conferences in Mathematical Logic, University of Connecticut, Starrs, Connecticut, 1979/80), Lecture Notes in Mathematics, vol. 859, Springer, Berlin, 1981, pp. 132–142.CrossRefGoogle Scholar
[Mar05] Marcone, Alberto, WQO and BQO theory in subsystems of second order arithmetic, Reverse mathematics (Simpson, S., editor). Lecture Notes in Logic, vol. 21, AK Peters, 2005, pp. 303–330.Google Scholar
[Mil85] Milner, E. C., 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.CrossRefGoogle Scholar
[Mon] Montalbán, Antonio, Equivalence between Fraïssé s conjecture and Jullien's theorem, To appear.Google Scholar
[Mon05] Montalbán, Antonio, Beyond the arithmetic, Ph.D. thesis, Cornell University, 2005, In preparation.Google Scholar
[Nas68] Nash-Williams, C. St. J. A., On better-quasi-ordering transfinite sequences, Proceedings of the Cambridge Philosophical Society, vol. 64 (1968), pp. 273–290.CrossRefGoogle Scholar
[Ros82] Rosenstein, Joseph, Linear orderings, Academic Press, New York – London, 1982.Google Scholar
[Sac90] Sacks, Gerald E., Higher recursion theory, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1990.CrossRefGoogle Scholar
[Sho93] Shore, Richard A., 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.Google Scholar
[Sim99] Simpson, Stephen G., Subsystems of second order arithmetic, Springer, 1999.CrossRefGoogle Scholar
[Spe55] Spector, Clifford, Recursive well-orderings, this Journal, vol. 20 (1955), pp. 151–163.Google Scholar