It is proved that the ordinal ω1cannot be embedded into a preordering Σ-definable with parameters in the hereditarily finite superstructure over the real numbers. As a corollary, we obtain the descriptions of ordinals Σ-presentable over\( \mathbb{H}\mathbbm{F} \)(ℝ) and of Gödel constructive sets of the form Lα. It is also shown that there are no Σ-presentations of structures of T-, m-, 1- and tt-degrees.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J. Barwise, Admissible Sets and Structures, Springer, Berlin (1975).
Yu. L. Ershov, Definability and Computability, Sib. School Alg. Log. [in Russian], Nauch. Kniga, Novosibirsk (1996).
H. Rogers, Theory of Recursive Functions and Effective Computability, McGraw-Hill, Maidenhead, Berksh. (1967).
G. E. Sacks, Higher Recursion Theory, Springer, Berlin (1990).
Yu. L. Ershov, V. G. Puzarenko, and A. I. Stukachev, “\( \mathbb{H}\mathbbm{F} \)-computability,” in Computability in Context. Computation and Logic in the Real World, S. B. Cooper et al. (Eds.), Imperial College Press, London (2011), pp. 169-242.
Yu. L. Ershov, “Σ-definability in admissible sets,” Dokl. Akad. Nauk SSSR, 285, No. 4, 792-795 (1985).
Yu. L. Ershov, “Σ-definability of algebraic structures,” in Handbook of Recursive Mathematics, Vol. 1, Recursive Model Theory, Y. L. Ershov, S. S. Goncharov, A. Nerode, and J. B. Remmel (eds.), Elsevier, Amsterdam (1998), pp. 235-260.
S. S. Goncharov and Yu. L. Ershov, Constructive Models, Siberian School of Algebra and Logic [in Russian], Nauch. Kniga, Novosibirsk (1999).
Yu. L. Ershov, Problems of Decidability and Constructive Models [in Russian], Nauka, Moscow (1980).
Yu. L. Ershov, “Definability in hereditarily finite superstructures,” Dokl. Ross. Akad. Nauk, 340, No. 1, 12-14 (1995).
A. S. Morozov and M. V. Korovina, “Σ-definability of countable structures over real numbers, complex numbers, and quaternions,” Algebra and Logic, 47, No. 3, 193-209 (2008).
A. S. Morozov, “Nonpresentability of some structures of analysis in hereditarily finite superstructures,” Algebra and Logic, 56, No. 6, 458-472 (2017).
A. S. Morozov, “A sufficient condition for nonpresentability of structures in hereditarily finite superstructures,” Algebra and Logic, 55, No. 3, 242-251 (2016).
A. N. Khisamiev, “Strong Δ1-definability of a model in an admissible set,” Sib. Math. J., 39, No. 1, 168-175 (1998).
A. V. Romina, “Definability of Boolean algebras in \( \mathbb{H}\mathbbm{F} \)-superstructures,” Algebra and Logic, 39, No. 6, 407-411 (2000).
A. N. Khisamiev, “On the Ershov upper semilattice LE,” Sib. Math. J., 45, No. 1, 173-187 (2004).
A. I. Stukachev, Σ-Definability in hereditarily finite superstructures and pairs of models, Algebra and Logic, 43, No. 4, 258-270 (2004).
A. I. Stukachev, “Degrees of presentability of structures. I,” Algebra and Logic, 46, No. 6, 419-432 (2007).
A. I. Stukachev, Σ-definability of uncountable models of c-simple theories, Sib. Math. J., 51, No. 3, 515-524 (2010).
A. Tarski, A Decision Method for Elementary Algebra and Geometry, Univ. Calif. Press, Berkeley (1951).
D. Marker, Model Theory: An Introduction, Grad. Texts Math., 217, Springer, New York (2002).
A. S. Morozov, “Some presentations of the real number field,” Algebra and Logic, 51, No. 1, 66-88 (2012).
B. L. Van der Waerden, Algebra. I, Springer, Berlin (1971).
L. Harrington and S. Shelah, “Counting equivalence classes for co-κ-Souslin equivalence relations,” Stud. Log. Found. Math., 108, 147-152 (1982).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Algebra i Logika, Vol. 58, No. 5, pp. 609-626, September-October, 2019.
Rights and permissions
About this article
Cite this article
Morozov, A.S. Σ-Preorderings in \( \mathbb{H}\mathbbm{F} \)(ℝ). Algebra Logic 58, 405–416 (2019). https://doi.org/10.1007/s10469-019-09560-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10469-019-09560-0