Power Kripke–Platek set theory and the axiom of choice

Abstract

Abstract While power Kripke–Platek set theory, ${\textbf{KP}}({\mathcal{P}})$, shares many properties with ordinary Kripke–Platek set theory, ${\textbf{KP}}$, in several ways it behaves quite differently from ${\textbf{KP}}$. This is perhaps most strikingly demonstrated by a result, due to Mathias, to the effect that adding the axiom of constructibility to ${\textbf{KP}}({\mathcal{P}})$ gives rise to a much stronger theory, whereas in the case of ${\textbf{KP}}$, the constructible hierarchy provides an inner model, so that ${\textbf{KP}}$ and ${\textbf{KP}}+V=L$ have the same strength. This paper will be concerned with the relationship between ${\textbf{KP}}({\mathcal{P}})$ and ${\textbf{KP}}({\mathcal{P}})$ plus the axiom of choice or even the global axiom of choice, $\textbf{AC}_{\tiny {global}}$. Since $L$ is the standard vehicle to furnish a model in which this axiom holds, the usual argument for demonstrating that the addition of ${\textbf{AC}}$ or $\textbf{AC}_{\tiny {global}}$ to ${\textbf{KP}}({\mathcal{P}})$ does not increase proof-theoretic strength does not apply in any obvious way. Among other tools, the paper uses techniques from ordinal analysis to show that ${\textbf{KP}}({\mathcal{P}})+\textbf{AC}_{\tiny {global}}$ has the same strength as ${\textbf{KP}}({\mathcal{P}})$, thereby answering a question of Mathias. Moreover, it is shown that ${\textbf{KP}}({\mathcal{P}})+\textbf{AC}_{\tiny {global}}$ is conservative over ${\textbf{KP}}({\mathcal{P}})$ for $\varPi ^1_4$ statements of analysis. The method of ordinal analysis for theories with power set was developed in an earlier paper. The technique allows one to compute witnessing information from infinitary proofs, providing bounds for the transfinite iterations of the power set operation that are provable in a theory. As the theory ${\textbf{KP}}({\mathcal{P}})+\textbf{AC}_{\tiny {global}}$ provides a very useful tool for defining models and realizability models of other theories that are hard to construct without access to a uniform selection mechanism, it is desirable to determine its exact proof-theoretic strength. This knowledge can for instance be used to determine the strength of Feferman’s operational set theory with power set operation as well as constructive Zermelo–Fraenkel set theory with the axiom of choice.