Nonpresentability of some structures of analysis in hereditarily finite superstructures

A. S. Morozov

Algebra i logika, Tome 56 (2017) no. 6, pp. 691-711.

Résumé

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.

Détail

Keywords: $\Sigma$-presentability, countable consistent theory, hereditarily finite superstructure, existentially Steinitz structure, semigroup of continuous functions, nonstandard analysis, infinite-dimensional separable Hilbert space.

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