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# Bounding prime models

## Abstract.

A set X is prime bounding if for every complete atomic decidable (CAD) theory T there is a prime model of T decidable in X. It is easy to see that X = 0′ is prime bounding. Denisov claimed that every X <T 0′ is not prime bounding, but we discovered this to be incorrect. Here we give the correct characterization that the prime bounding sets Xτ 0′ are exactly the sets which are not low2. Recall that X is low2 if X″ ≤τ 0″. To prove that a low2 set X is not prime bounding we use a 0′ -computable listing of the array of sets {Y : YτX } to build a CAD theory T which diagonalizes against all potential X-decidable prime models of T, To prove that any non-low2X is indeed prime bounding. we fix a function fTX that is not dominated by a certain 0′-computable function that picks out generators of principal types. Given a CAD theory T. we use f to eventually find, for every formula φ() con sistent with T. a principal type which contains it. and hence to build an X-decidable prime model of T. We prove the prime bounding property equivalent to several other combinatorial properties, including some related to the limitwise monotonic functions which have been introduced elsewhere in computable model theory.

## References

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[1]Ash, C. J. and Knight, J. F., Computable structures and the hyperarithmetical hierarchy, Elsevier Science, Amsterdam, 2000.
[2]Chang, C. C. and Keisler, H. J., Model theory, 3rd ed., Studies in Logic and the Foundations of Mathematics, vol. 73, North-Holland, Amsterdam, 1990, 1st ed. 1973, 2nd ed. 1977.
[3]Coles, R. J., Downey, R., and Khoussainov, B., On initial segments of computable linear orders, Order, vol. 14 (1998), pp. 107–124.
[4]Csima, B. F., Degree spectra of prime models, this Journal, vol. 69 (2004), pp. 430–442.
[5]Csima, B. F., Harizanov, V. S., Hirschfeldt, D. R., and Soare, R. I., Bounding homogeneous models, to appear.
[6]Denisov, A. S., Homogeneous 0′-elements in structural pre-orders, Algebra and Logic, vol. 28 (1989), pp. 405–418.
[7]Drobotun, B. N.. Enumerations of simple models. Siberian Mathematics Journal, vol. 18 (1978), pp. 707–716.
[8]Goncharov, S. S. and Nurtazin, A. T., Constructive models of complete decidable theories, Algebra and Logic, vol. 12 (1973), pp. 67–77.
[9]Harizanov, V. S., Pure computable model theory, Handbook of recursive mathematics (Ershov, Yu. L., Goncharov, S. S., Nerode, A., and Remmel, J. B., editors), Studies in Logic and the Foundations of Mathematics, vol. 138–139, Elsevier Science, Amsterdam, 1998, pp. 3–114.
[10]Harizanov, V. S., Computability-theoretic complexity of countable structures, The Bulletin of Symbolic Logic, vol. 8 (2002), pp. 457–477.
[11]Harrington, L., Recursively presentable prime models, this Journal, vol. 39 (1974), pp. 305–309.
[12]Hirschfeldt, D. R., Prime models of theories of computable linear orderings, Proceedings of the American Mathematical Society, vol. 129 (2001), pp. 3079–3083.
[13]Jockusch, C. G. Jr., Degrees in which the recursive sets are uniformly recursive, Canadian Journal of Mathematics, vol. 24 (1972), pp. 1092–1099.
[14]Jockusch, C. G. Jr. and Soare, R. I., classes and degrees of theories, Transactions of the American Mathematical Society, vol. 173 (1972), pp. 33–56.
[15]Kaplansky, I., Infinite Abelian groups, University of Michigan Press, 1954.
[16]Khisamiev, N. G., A constructibility criterion for the direct product of cyclic p-groups, Izvestiya Akademiya Nauk Kazakhstan SSR, Seriya Fiziko-Matematicheskaya, (1981), pp. 51–55, in Russian.
[17]Khisamiev, N. G., Theory of abelian groups with constructive models, Siberian Mathematics Journal, vol. 27 (1986), pp. 572–585.
[18]Khisamiev, N. G., Constructive abelian groups, Handbook of recursive mathematics (Ershov, Yu. L., Goncharov, S. S., Nerode, A., and Remmel, J. B., editors), Studies in Logic and the Foundations of Mathematics, vol. 138–139, Elsevier Science, Amsterdam, 1998, pp. 1177–1231.
[19]Khoussainov, B., Nies, A., and Shore, R. A., Computable models of theories with few models, Notre Dame Journal of Formal Logic, vol. 38 (1997), pp. 165–178.
[20]Knight, J. F., Degrees coded in jumps of orderings, this Journal, vol. 51 (1986), pp. 1034–1042.
[21]Knight, J. F., Degrees of models, Handbook of recursive mathematics (Ershov, Yu. L., Goncharov, S. S., Nerode, A., and Remmel, J. B., editors), Studies in Logic and the Foundations of Mathematics, vol. 138–139, Elsevier Science, Amsterdam, 1998, pp. 289–309.
[22]Martin, D. A., Classes of recursively enumerable sets and degrees of unsolvability, Zeitschrift für mathematische Logik unddie Grundlagen der Mathematik, vol. 12 (1966), pp. 295–310.
[23]Millar, T. S., Foundations of recursive model theory, Annals of Mathematical Logic, vol. 13 (1978), pp. 45–72.
[24]Millar, T. S., Omitting types, type spectrums, and decidability, this Journal, vol. 48 (1983), pp. 171–181.
[25]Nies, A., A new spectrum of recursive models, Notre Dame Journal of Formal Logic, vol. 40 (1999), pp. 307–314.
[26]Sacks, G. E., Saturated model theory, W. A. Benjamin, Inc., Reading, MA, 1972.
[27]Soare, R. I., Recursively enumerable sets and degrees: A study of computable functions and computably generated sets, Springer-Verlag, Heidelberg, 1987.

# Bounding prime models

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