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CLASSES OF STRUCTURES WITH NO INTERMEDIATE ISOMORPHISM PROBLEMS

  • ANTONIO MONTALBÁN (a1)

Abstract

We say that a theory T is intermediate under effective reducibility if the isomorphism problems among its computable models is neither hyperarithmetic nor on top under effective reducibility. We prove that if an infinitary sentence T is uniformly effectively dense, a property we define in the paper, then no extension of it is intermediate, at least when relativized to every oracle in a cone. As an application we show that no infinitary sentence whose models are all linear orderings is intermediate under effective reducibility relative to every oracle in a cone.

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[1]Ash, C. J. and Knight, J., Computable Structures and the Hyperarithmetical Hierarchy, Studies in Logic and the Foundations of Mathematics, vol. 144, Elsevier, Amsterdam, 2000.
[2]Barwise, Jon, Admissible Sets and Structures: An Approach to Definability Theory, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1975.
[3]Becker, Howard, Isomorphism of computable structures and Vaught’s conjecture, this Journal, vol. 78 (2013), no. 4, pp. 13281344.
[4]Becker, Howard and Kechris, Alexander S., The Descriptive Set Theory of Polish Group Actions, London Mathematical Society Lecture Note Series, vol. 232, Cambridge University Press, Cambridge, 1996.
[5]Burgess, John P., A reflection phenomenon in descriptive set theory. Fundamenta Mathematicae, vol. 104 (1979), no. 2, pp. 127139.
[6]Downey, Rod and Montalbán, Antonio, The isomorphism problem for torsion-free abelian groups is analytic complete. Journal of Algebra, vol. 320 (2008), pp. 22912300.
[7]Fokina, Ekaterina B. and Friedman, Sy-David, Equivalence relations on classes of computable structures, Mathematical Theory and Computational Practice, Lecture Notes in Computer Science, vol. 5635, Springer, Berlin, 2009, pp. 198207.
[8]Fokina, E. B., Friedman, S., Harizanov, V., Knight, J. F., McCoy, C., and Montalbán, A., Isomorphism and bi-embeddability relations on computable structures, this Journal, vol. 77 (2012), no. 1, pp. 122132.
[9]Friedman, Harvey and Stanley, Lee, A Borel reducibility theory for classes of countable structures, this Journal, vol. 54 (1989), no. 3, pp. 894914.
[10]Gao, Su, Some dichotomy theorems for isomorphism relations of countable models, this Journal, vol. 66 (2001), no. 2, pp. 902922.
[11]Gao, Su, Invariant descriptive set theory, Pure and Applied Mathematics, vol. 293, CRC Press, Boca Raton, FL, 2009.
[12]Harrison, J., Recursive pseudo-well-orderings. Transactions of the American Mathematical Society, vol. 131 (1968), pp. 526543.
[13]Hjorth, Greg, The isomorphism relation on countable torsion free abelian groups. Fundamenta Mathematicae, vol. 175 (2002), no. 3, pp. 241257.
[14]Harnik, V. and Makkai, M., A tree argument in infinitary model theory. Proceedings of the American Mathematical Society, vol. 67 (1977), no. 2, pp. 309314.
[15]Kanamori, Akihiro, The Higher Infinite, second ed., Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003.
[16]Knight, Julia and Montalbán, Antonio, $\sum _1^1 $-equivalence relations which are on top, unpublished notes, September 2010.
[17]Montalbán, Antonio, On the equimorphism types of linear orderings. Bulletin of Symbolic Logic, vol. 13 (2007), no. 1, pp. 7199.
[18]Montalbán, Antonio, A computability theoretic equivalent to Vaught’s conjecture. Advances in Mathematics, vol. 235 (2013), pp. 5673.
[19]Montalbán, Antonio, Priority arguments via true stages, this Journal, accepted.
[20]Montalbán, Antonio, Analytic equivalence relations satisfying hyperarithmetic-is-recursive, submitted.
[21]Nadel, Mark, Scott sentences and admissible sets. Annals of Mathematical Logic, vol. 7 (1974), pp. 267294.
[22]Rosenstein, Joseph G., Linear orderings, Pure and Applied Mathematics, vol. 98, Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1982.
[23]Rubin, Matatyahu, Theories of linear order. Israel Journal of Mathematics, vol. 17 (1974), pp. 392443.
[24]Sacks, Gerald E., Countable admissible ordinals and hyperdegrees. Advances in Mathematics, vol. 20 (1976), no. 2, pp. 213262.
[25]Sacks, Gerald E., Bounds on weak scattering. Notre Dame Journal of Formal Logic, vol. 48 (2007), no. 1, pp. 531.
[26]Silver, Jack H., Counting the number of equivalence classes of Borel and coanalytic equivalence relations. Annals of Mathematical Logic, vol. 18 (1980), no. 1, pp. 128.
[27]Slaman, Theodore A. and Steel, John R., Definable functions on degrees, Cabal Seminar 81–85, Lecture Notes in Mathematics, vol. 1333, Springer, Berlin, 1988, pp. 3755.
[28]Steel, John R., On Vaught’s conjecture, Cabal Seminar 76–77 (Proc. Caltech-UCLA Logic Sem., 1976–77), Lecture Notes in Mathematics, vol. 689, Springer, Berlin, 1978, pp. 193208.
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