Elementary Extensions of External Classes in a Nonstandard Universe

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In continuation of our study of HST, Hrbaček set theory (a nonstandard set theory which includes, in particular, the ZFC Replacement and Separation schemata in the st-ε-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 κ, any set R ⊆ *κ generates an “internal” class 𝒮(R) of all sets standard relatively to elements of R, and an “external” class ℒ[𝒮;(R)] of all sets constructible (in a sense close to the Gödel constructibility) from sets in 𝒮(R). We prove that under some mild saturation-like requirements for R the class ℒ[𝒮(R)] models a certain κ-version of HST including the principle of κ+-saturation; moreover, in this case ℒ;[𝒮(R′)] is an elementary extension of ℒ[𝒮;(R)] in the st-ε-language whenever sets R ⊆ R′ satisfy the requirements.

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