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

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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/}
}
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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/