Nonpresentability of some structures of analysis in hereditarily finite superstructures

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.