\(\Sigma\)-presentations of the ordering on the reals. (English. Russian original) Zbl 1323.03047
Algebra Logic 53, No. 3, 217-237 (2014); translation from Algebra Logika 53, No. 3, 340-371 (2014).
Summary: We prove the nonexistence of universal \(\Sigma\)-presentable linear orderings as well as the effective infinity of the class of \(\Sigma\)-presentations of the natural order on \(\mathbb{R}\) over an admissible set \(\mathbb{HF} (\mathbb{R})\).
MSC:
| 03C57 | Computable structure theory, computable model theory |
| 03D45 | Theory of numerations, effectively presented structures |
| 03C70 | Logic on admissible sets |
References:
| [1] | A. I. Mal’tsev, “On recursive Abelian groups,” Dokl. Akad. Nauk SSSR, 146, No. 5, 1009-1012 (1962). · Zbl 0156.01105 |
| [2] | S. S. Goncharov, “The problem of the number of non-autoequivalent constructivizations,” Dokl. Akad. Nauk SSSR, 251, No. 2, 271-274 (1980). · Zbl 0476.03045 |
| [3] | S. S. Goncharov, “The problem of the number of non-autoequivalent constructivizations,” Algebra and Logic, 19, No. 6, 401-414 (1980). · Zbl 0476.03046 · doi:10.1007/BF01669323 |
| [4] | D. R. Hirschfeldt, B. Khoussainov, R. A. Shore, and A. M. Slinko, “Degree spectra and computable dimension in algebraic structures,” Ann. Pure Appl. Log., 115, Nos. 1-3, 71-113 (2002). · Zbl 1016.03034 · doi:10.1016/S0168-0072(01)00087-2 |
| [5] | Ershov, YL; Ershov, YL (ed.); Goncharov, SS (ed.); Nerode, A. (ed.); Remmel, JB (ed.), Σ-definability of algebraic structures, 235-260 (1998), Amsterdam · Zbl 0940.03043 · doi:10.1016/S0049-237X(98)80006-2 |
| [6] | 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). · Zbl 1164.03332 · doi:10.1007/s10469-008-9009-x |
| [7] | A. V. Romina, “Autostability of hyperarithmetical models,” Algebra and Logic, 39, No. 2, 114-118 (2000). · Zbl 0953.03044 · doi:10.1007/BF02681665 |
| [8] | J. Barwise, Admissible Sets and Structures, Springer, Berlin (1975). · Zbl 0316.02047 · doi:10.1007/978-3-662-11035-5 |
| [9] | Yu. L. Ershov, Definability and Computability, Sib. School Alg. Log. [in Russian], Nauch. Kniga, Novosibirsk (1996). · Zbl 0856.03039 |
| [10] | A. Tarski, A Decision Method for Elementary Algebra and Geometry, Univ. Calif. Press, Berkeley (1951). · Zbl 0044.25102 |
| [11] | D. Marker, Model Theory: An Introduction, Grad. Texts Math., 217, Springer, New York (2002). · Zbl 1003.03034 |
| [12] | Ershov, YL; Puzarenko, VG; Stukachev, AI; Cooper, SB (ed.); Sorbi, A. (ed.), HF-computability, 173-248 (2011), London |
| [13] | A. S. Morozov, “Some presentations of the real number field,” Algebra and Logic, 51, No. 1, 66-88 (2012). · Zbl 1334.03039 · doi:10.1007/s10469-012-9171-z |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
