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About the Strength of Operational Regularity

We analyze the consistency strength of regularity on the basis of Feferman’s operational set theory OST. Our main result shows that regularity over OST for operations corresponds to regularity with respect to set-theoretic functions in frameworks like Kripke-Platek set theory. As a consequence, we obtain that OST plus the operational axiom (Inac) is consistency-equivalent to KripkePlatek set theory with infinity extended by the strong limit axiom (SLim) stating that any ordinal is majorized by a functionally regular ordinal. Thus OST+ (Inac) is significantly stronger than originally expected.