We construct a decidable prime model in which the degree of a set of complete formulas is equal to 0', infinitely many tuples of elements comply with every complete formula, and the decidable categoricity spectrum coincides with the set of all PA-degrees.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
S. S. Goncharov, “Degrees of autostability relative to strong constructivizations,” Trudy MIAN, 274, 119-129 (2011).
A. T. Nurtazin, “Strong and weak constructivizations and computable families,” Algebra and Logic, 13, No. 3, 177-184 (1974).
N. A. Bazhenov, “Degrees of autostability relative to strong constructivizations for Boolean algebras,” Algebra and Logic, 55, No. 2, 87-102 (2016).
N. Bazhenov and M. I. Marchuk, “Degrees of autostability relative to strong constructivizations of graphs,” Sib. Math. J., 59, No. 4, 565-577 (2018).
S. S. Goncharov, “Autostability of prime models with respect to strong constructivizations,” Algebra and Logic, 48, No. 6, 410-417 (2009).
S. S. Goncharov, “On the autostability of almost prime models with respect to strong constructivizations,” Usp. Mat. Nauk, 65, No. 5(395), 107-142 (2010).
N. Bazhenov, “Autostability spectra for decidable structures,” Math. Struct. Comput. Sci., 28, No. 3, 392-411 (2018).
S. S. Goncharov, N. A. Bazhenov, and M. I. Marchuk, “The index set of Boolean algebras autostable relative to strong constructivizations,” Sib. Math. J., 56, No. 3, 394–404 (2015).
S. S. Goncharov, V. Harizanov, and R. Miller, “On decidable categoricity and almost prime models,” Sib. Adv. Math., 30, No. 3, 200-212 (2020).
N. A. Bazhenov, “Autostability spectra for Boolean algebras,” Algebra and Logic, 53, No. 6, 502-505 (2014).
N. A. Bazhenov, I. Sh. Kalimullin, and M. M. Yamaleev, “Degrees of categoricity vs. strong degrees of categoricity,” Algebra and Logic, 55, No. 2, 173-177 (2016).
B. F. Csima, J. N. Franklin, and R. A. Shore, “Degrees of categoricity and the hyperarithmetic hierarchy,” Notre Dame J. Form. Log., 54, No. 2, 215-231 (2013).
E. Fokina, A. Frolov, and I. Kalimullin, “Categoricity spectra for rigid structures,” Notre Dame J. Form. Log., 57, No. 1, 45-57 (2016).
E. B. Fokina, I. Kalimullin, and R. Miller, “Degrees of categoricity of computable structures,” Arch. Math. Log., 49, No. 1, 51-67 (2010).
R. Miller and A. Shlapentokh, “Computable categoricity for algebraic fields with splitting algorithms,” Trans. Am. Math. Soc., 367, No. 6, 3955-3980 (2015).
C. C. Chang and H. J. Keisler, Model Theory, North-Holland, Amsterdam (1973).
D. Scott, “Algebras of sets binumerable in complete extensions of arithmetic,” Proc. Symp. Pure Math., 5, 117-121 (1962).
C. G. Jockusch and R. I. Soare, “\( {\Pi}_1^0 \) classes and degrees of theories,” Trans. Am. Math. Soc., 173, 33-56 (1972).
S. G. Simpson, “Degrees of unsolvability: A survey of results,” in Handbook of Mathematical Logic, Stud. Log. Found. Math., 90, J. Barwise (ed.), North-Holland, Amsterdam (1977), pp. 631-652.
M. Harrison-Trainor, “There is no classification of the decidably presentable structures,” J. Math. Log., 18, No. 2 (2018), Article ID 1850010.
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by RFBR, project No. 20-01-00300.
Supported byMathematical Center in Akademgorodok, Agreement with RFMinistry of Education and Science No. 075-15-2019-1613.
Translated from Algebra i Logika, Vol. 60, No. 3, pp. 303-312, May-June, 2021. Russian DOI: https://doi.org/10.33048/alglog.2021.60.304.
Rights and permissions
About this article
Cite this article
Goncharov, S.S., Marchuk, M.I. The Degree of Decidable Categoricity of a Model with Infinite Solutions for Complete Formulas. Algebra Logic 60, 200–206 (2021). https://doi.org/10.1007/s10469-021-09642-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10469-021-09642-y