Classifying the provably total set functions of KP and KP(P)

Abstract

This article is concerned with classifying the provably total set-functions of Kripke-Platek set theory, KP, and Power Kripke-Platek set theory, KP(P), as well as proving several (partial) conservativity results. The main technical tool used in this paper is a relativisation technique where ordinal analysis is carried out relative to an arbitrary but fixed set x. A classic result from ordinal analysis is the characterisation of the provably recursive functions of Peano Arithmetic, PA, by means of the fast growing hierarchy [10]. Whilst it is possible to formulate the natural numbers within KP, the theory speaks primarily about sets. For this reason it is desirable to obtain a characterisation of its provably total set functions. We will show that KP proves the totality of a set function precisely when it falls within a hierarchy of set functions based upon a relativised constructible hierarchy stretching up in length to any ordinal below the Bachmann-Howard ordinal. As a consequence of this result we obtain that IKP + forall x,y (x in y or x not in y) is conservative over KP for forall-exists-formulae, where IKP stands for intuitionistic Kripke-Platek set theory. In a similar vein, utilising [56], it is shown that KP(P) proves the totality of a set function precisely when it falls within a hierarchy of set functions based upon a relativised von Neumann hierarchy of the same length. The relativisation technique applied to KP(P) with the global axiom of choice, AC-global, also yields a parameterised extension of a result in [58], showing that KP(P) + AC-global is conservative over KP(P) + AC and CZF + AC for Pi_2 in powerset statements. Here AC stands for the ordinary axiom of choice and CZF refers to constructive Zermelo-Fraenkel set theory.