Nonpresentability of Some Structures of Analysis in Hereditarily Finite Superstructures

Abstract

It is proved that any countable consistent theory with infinite models has a Σ-presentable model of cardinality 2^ω over ℍ????(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 Σ-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.