Conservativity of SPOTS over ZF

Determine whether the multi-level nonstandard set theory SPOTS is a conservative extension of ZF, i.e., whether every theorem in the E-language provable in SPOTS is already provable in ZF.

Background

The paper introduces SPOTS and SCOTS, extending earlier single-level nonstandard set theories (SPOT and SCOT) to frameworks with many levels of standardness. A central theme is minimizing reliance on strong forms of the Axiom of Choice while retaining nonstandard analytical power.

The authors establish conservativity results: SCOTS over ZF + ADC and SPOTS over ZF + ACC. However, the full conservativity of SPOTS over ZF (without any choice) remains unresolved, and settling this would clarify the precise strength of SPOTS relative to ZF.