Geodesic
Nonpresentability of some structures of analysis in hereditarily finite superstructures
Algebra i logika, Tome 56 (2017) no. 6, pp. 691-711

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It is proved that any countable consistent theory with infinite models has a $\Sigma$-presentable model of cardinality $2^\omega$ over $\mathbb{HF(R})$. It is shown that some structures studied in analysis (in particular, a semigroup of continuous functions, certain structures of nonstandard analysis, and infinite-dimensional separable Hilbert spaces) have no simple $\Sigma$-presentations in hereditarily finite superstructures over existentially Steinitz structures. The results are proved by a unified method on the basis of a new general sufficient condition.
Keywords: $\Sigma$-presentability, countable consistent theory, hereditarily finite superstructure, existentially Steinitz structure, semigroup of continuous functions, nonstandard analysis, infinite-dimensional separable Hilbert space. 
@article{AL_2017_56_6_a3,
     author = {A. S. Morozov},
     title = {Nonpresentability of some structures of analysis in hereditarily finite superstructures},
     journal = {Algebra i logika},
     pages = {691--711},
     publisher = {mathdoc},
     volume = {56},
     number = {6},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/AL_2017_56_6_a3/}
}
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A. S. Morozov. Nonpresentability of some structures of analysis in hereditarily finite superstructures. Algebra i logika, Tome 56 (2017) no. 6, pp. 691-711. http://geodesic.mathdoc.fr/item/AL_2017_56_6_a3/