The paper relativizes the method of ordinal analysis developed for Kripke–Platek set theory to theories which have the power set axiom. We show that it is possible to use this technique to extract information about Power Kripke–Platek set theory, KP.As an application it is shown that whenever KP+AC proves a ΠP2 statement then it holds true in the segment Vτ of the von Neumann hierarchy, where τ stands for the Bachmann–Howard ordinal.

Most axiomatizations of set theory that have been treated metamathematically have been based either entirely on classical logic or entirely on intuitionistic logic. But a natural conception of the settheoretic universe is as an indefinite (or “potential”) totality, to which intuitionistic logic is more appropriately applied, while each set is taken to be a definite (or “completed”) totality, for which classical logic is appropriate; so on that view, set theory should be axiomatized on some correspondingly mixed basis. Similarly, in the (...) case of predicative analysis, the natural numbers are considered to form a definite totality, while the universe of sets (or functions) of natural numbers are viewed as an indefinite totality, so that, again, a mixed semi-constructive logic should be the appropriate one to treat the two together. Various such semiconstructive systems of analysis and set theory are formulated here and their proof-theoretic strength is characterized. Interestingly, though the logic is weakened, one can in compensation strengthen certain principles in a way that could be advantageous for mathematical applications. (shrink)