Elementary extensions of external classes in a nonstandard universe

In continuation of our study of HST, Hrbacek set theory (a nonstandard set theory which includes, in particular, the ZFC Replacement and Separation schemata in the st-6-language, and Saturation for well-orderable families of internal sets), we consider the problem of existence of elementary extensions of inner "external" subclasses of the HST universe. We show that, given a standard cardinal K, any set R ⊂K generates an "internal" class (ℛ) of all sets standard relatively to elements of R, and an "external" class [(.ℛ)] of all sets constructible (in a sense close to the Gödel constructibility) from sets in (ℛ). We prove that under some mild saturation-like requirements for R the class [(ℛ)] models a certain k-version of HST including the principle of K -saturation; moreover, in this case [(ℛ')] is an elementary extension of L[S(.R)] in the st-∈-language whenever sets R ⊂ R' satisfy the requirements. Key words:. © 1998 Kluwer Academic Publishers.