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Degrees of categoricity of computable structures. (English) Zbl 1184.03026
Summary: Defining the degree of categoricity of a computable structure \({\mathcal{M}}\) to be the least degree \(\mathbf d\) for which \({\mathcal{M}}\) is \(\mathbf d\)-computably categorical, we investigate which Turing degrees can be realized as degrees of categoricity. We show that, for all \(n\), degrees d.c.e. in and above \(\mathbf 0^{(n)}\) can be so realized, as can the degree \(\mathbf 0^{(\omega)}\).

MSC:
03C57 Computable structure theory, computable model theory
03C35 Categoricity and completeness of theories
03D28 Other Turing degree structures
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[1] Cholak P., Goncharov S.S., Khoussainov B., Shore R.A.: Computably categorical structures and expansions by constants. J. Symb. Log. 64, 13–37 (1999) · Zbl 0928.03040 · doi:10.2307/2586747
[2] Fröhlich A., Shepherdson J.C.: Effective procedures in field theory. Philos. Trans. R. Soc. Lond. A 248(950), 407–432 (1956) · Zbl 0070.03502 · doi:10.1098/rsta.1956.0003
[3] Goncharov S.S.: Problem of the number of non-self-equivalent constructivizations. Algebra Log. 19, 401–414 (1980) · Zbl 0476.03046 · doi:10.1007/BF01669323
[4] Goncharov, S.S.: Nonequivalent constructivizations. In: Proceedings of Mathematical Institute of the Siberian Branch Academy of Sciences, Nauka, Novosibirsk (1982) · Zbl 0543.03017
[5] Goncharov, S.S.: Autostable models and algorithmic dimensions. In: Ershov, Y.L., Goncharov, S.S., Nerode, A., Remmel, J.B. (eds.) Handbook of Recursive Mathematics, vol. 1, pp. 261–287. Elsevier, Amsterdam (1998) · Zbl 0958.03030
[6] Goncharov S.S., Khoussainov B.: Complexity of categorical theories with computable models. Algebra Log. 43(6), 365–373 (2004) · Zbl 1097.03027 · doi:10.1023/B:ALLO.0000048826.92325.02
[7] Goncharov S., Harizanov V., Knight J., McCoy C., Miller R., Solomon R.: Enumerations in computable structure theory. Ann. Pure Appl. Log. 136(3), 219–246 (2005) · Zbl 1081.03033 · doi:10.1016/j.apal.2005.02.001
[8] Harizanov V.S.: Pure computable model theory. In: Ershov, Y.L., Goncharov, S.S., Nerode, A., Remmel, J.B. (eds) Handbook of Recursive Mathematics, vol. 1, pp. 3–114. Elsevier, Amsterdam (1998) · Zbl 0952.03037
[9] Harizanov, V.S., Miller, R., Morozov, A.: Automorphism spectra of computable structures (to appear)
[10] Harrison, J.: Doctoral dissertation, Stanford University (1967)
[11] Hirschfeldt D.R., Khoussainov B., Shore R.A., Slinko A.M.: Degree spectra and computable dimensions in algebraic structures. Ann. Pure Appl. Log. 115, 71–113 (2002) · Zbl 1016.03034 · doi:10.1016/S0168-0072(01)00087-2
[12] Kreisel G., Shoenfield J., Wang H.: Number theoretic concepts and recursive well-orderings. Arch. Math. Log. Grundl. 5, 42–64 (1960) · Zbl 0129.00402 · doi:10.1007/BF01977642
[13] Marker D.: Non-\(\Sigma\) n -axiomatizable almost strongly minimal theories. J. Symb. Log. 54, 921–927 (1989) · Zbl 0698.03021 · doi:10.2307/2274752
[14] Miller R.: d-Computable categoricity for algebraic fields. J. Symb. Log. 74(4), 1325–1351 (2009) · Zbl 1202.03044 · doi:10.2178/jsl/1254748694
[15] Moschovakis Y.N.: Descriptive Set Theory, vol. 100 of Studies in Logic and the Foundations of Mathematics. North-Holland, Amsterdam (1980) · Zbl 0517.03025
[16] Richter L.J.: Degrees of structures. J. Symb. Log. 46, 723–731 (1981) · Zbl 0512.03024 · doi:10.2307/2273222
[17] Soare R.I.: Recursively Enumerable Sets and Degrees. Springer, New York (1987) · Zbl 0667.03030
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