Geodesic
$\Sigma$-preorderings in ${\mathbb{HF}(\mathbb{R})}$
Algebra i logika, Tome 58 (2019) no. 5, pp. 609-626.

Voir la notice de l'article provenant de la source Math-Net.Ru

It is proved that the ordinal $\omega_1$ cannot be embedded into a preordering $\Sigma$-definable with parameters in the hereditarily finite superstructure over the real numbers. As a corollary, we obtain the descriptions of ordinals $\Sigma$-presentable over ${\mathbb{HF}(\mathbb{R})}$ and of Gödel constructive sets of the form $L_\alpha$. It is also shown that there are no $\Sigma$-presentations of structures of $T$-, $m$-, $1$- and $tt$-degrees.
Keywords: $\Sigma$-definable preordering, ordinal, hereditarily finite superstructure, real numbers. 
@article{AL_2019_58_5_a2,
     author = {A. S. Morozov},
     title = {$\Sigma$-preorderings in ${\mathbb{HF}(\mathbb{R})}$},
     journal = {Algebra i logika},
     pages = {609--626},
     publisher = {mathdoc},
     volume = {58},
     number = {5},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/AL_2019_58_5_a2/}
}
TY  - JOUR
AU  - A. S. Morozov
TI  - $\Sigma$-preorderings in ${\mathbb{HF}(\mathbb{R})}$
JO  - Algebra i logika
PY  - 2019
SP  - 609
EP  - 626
VL  - 58
IS  - 5
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/AL_2019_58_5_a2/
LA  - ru
ID  - AL_2019_58_5_a2
ER  - 
%0 Journal Article
%A A. S. Morozov
%T $\Sigma$-preorderings in ${\mathbb{HF}(\mathbb{R})}$
%J Algebra i logika
%D 2019
%P 609-626
%V 58
%N 5
%I mathdoc
%U http://geodesic.mathdoc.fr/item/AL_2019_58_5_a2/
%G ru
%F AL_2019_58_5_a2
A. S. Morozov. $\Sigma$-preorderings in ${\mathbb{HF}(\mathbb{R})}$. Algebra i logika, Tome 58 (2019) no. 5, pp. 609-626. http://geodesic.mathdoc.fr/item/AL_2019_58_5_a2/

[1] J. Barwise, Admissible sets and structures. An approach to definability theory, Perspec. Math. Logic, Springer-Velag, Berlin, 1975 | DOI | MR | Zbl

[2] Yu. L. Ershov, Opredelimost i vychislimost, Sibirskaya shkola algebry i logiki, Nauchnaya kniga (NII MIOO NGU), Novosibirsk, 1996 | MR

[3] H. Rogers, Theory of recursive functions and effective computability, McGraw-Hill, New York, 1967 ; Kh. Rodzhers, Teoriya rekursivnykh funktsii i effektivnaya vychislimost, Mir, M., 1972 | MR | Zbl

[4] G. E. Sacks, Higher recursion theory, Perspect. Math. Log., Springer-Verlag, Berlin etc, 1990 | DOI | MR | Zbl

[5] Yu. L. Ershov, V. G. Puzarenko, A. I. Stukachev, “${\mathbb {HF}}$-Computability”, Computability in context. Computation and logic in the real world, eds. S. B. Cooper et al., Imperial College Press, London, 2011, 169–242 | DOI | MR | Zbl

[6] Yu. L. Ershov, “$\Sigma$-opredelimost v dopustimykh mnozhestvakh”, Dokl. AN SSSR, 285:4 (1985), 792–795 | MR | Zbl

[7] Yu. L. Ershov, “$\Sigma$-definability of algebraic structures”, Handbook of recursive mathematics, v. 1, Stud. Logic Found. Math., 138, Recursive model theory, eds. Y. L. Ershov et al., Elsevier Science B.V., Amsterdam, 1998, 235–260 | DOI | MR | Zbl

[8] S. S. Goncharov, Yu. L. Ershov, Konstruktivnye modeli, Sibirskaya shkola algebry i logiki, Nauchnaya kniga, Novosibirsk, 1999 | MR

[9] Yu. L. Ershov, Problemy razreshimosti i konstruktivnye modeli, Nauka, M., 1980

[10] Yu. L. Ershov, “Opredelimost v nasledstvenno konechnykh nadstroikakh”, Dokl. RAN, 340:1 (1995), 12–14 | MR | Zbl

[11] A. S. Morozov, M. V. Korovina, “O $\Sigma$-opredelimosti schetnykh struktur nad veschestvennymi, kompleksnymi chislami i kvaternionami”, Algebra i logika, 47:3 (2008), 335–363 | MR | Zbl

[12] A. S. Morozov, “Nepredstavimost nekotorykh struktur analiza v nasledstvenno konechnykh nadstroikakh”, Algebra i logika, 56:6 (2017), 691–711 | MR

[13] A. S. Morozov, “Ob odnom dostatochnom uslovii nepredstavimosti struktur v nasledstvenno konechnykh nadstroikakh”, Algebra i logika, 55:3 (2016), 366–379 | MR | Zbl

[14] A. N. Khisamiev, “Silnaya $\Delta_1$-opredelimost modeli v dopustimom mnozhestve”, Sib. matem. zh., 39:1 (1998), 191–200 | MR | Zbl

[15] A. V. Romina, “Opredelimost bulevykh algebr v ${\mathbb{HF}}$-nadstroikakh”, Algebra i logika, 39:6 (2000), 711–719 | MR | Zbl

[16] A. N. Khisamiev, “O verkhnei polureshetke Ershova ${\mathfrak L}_E$”, Sib. matem. zh., 45:1 (2004), 211–228 | MR | Zbl

[17] A. I. Stukachev, “$\Sigma$-opredelimost v nasledstvenno konechnykh nadstroikakh i pary modelei”, Algebra i logika, 43:4 (2004), 459–481 | MR | Zbl

[18] A. I. Stukachev, “O stepenyakh predstavimosti modelei. I”, Algebra i logika, 46:6 (2007), 763–788 | MR | Zbl

[19] A. I. Stukachev, “$\Sigma$-Opredelimost neschetnykh modelei $c$-prostykh teorii”, Sib. matem. zh., 51:3 (2010), 649–661 | MR | Zbl

[20] A. Tarski, A decision method for elementary algebra and geometry, 2nd ed, Univ. Calif. Press, Berkeley, 1951 | MR | Zbl

[21] D. Marker, Model theory: an introduction, Grad. Texts Math., 217, Springer-Verlag, New York etc., 2002 | MR | Zbl

[22] A. S. Morozov, “O nekotorykh predstavleniyakh polya veschestvennykh chisel”, Algebra i logika, 51:1 (2012), 96–128 | MR | Zbl

[23] B. L. Van der Varden, Algebra, Nauka, M., 1976 | MR

[24] L. Harrington, S. Shelah, “Counting equivalence classes for co-$\kappa$-Souslin equivalence relations”, Logic colloquium '80, Eur. Summer Meet. (Prague, 1980), Stud. Logic Found. Math., 108, 1982, 147–152 | DOI | MR | Zbl