Admissible extensions of subtheories of second order arithmetic

Abstract

In this paper we study admissible extensions of several theories T of reverse mathematics. The idea is that in such an extension the structure $M=(N,S,\in )$ of the natural numbers and collection of sets of natural numbers $S$ has to obey the axioms of T while simultaneously one also has a set-theoretic world with transfinite levels erected on top of $M$ governed by the axioms of Kripke-Platek set theory, $\mathrm{KP}$.

In some respects, the admissible extension of T can be viewed as a proof-theoretic analog of Barwise's admissible cover of an arbitrary model of set theory; see [2]. However, by contrast, the admissible extension of T is usually not a conservative extension of T. Owing to the interplay of T and $\mathrm{KP}$, either theory's axioms may force new sets of naturals to exist which in turn may engender yet new sets of naturals on account of the axioms of the other.

The paper discerns a general pattern though. It turns out that for many familiar theories T, the second order part of the admissible cover of T equates to T augmented by transfinite induction over all initial segments of the Bachmann-Howard ordinal. Technically, the paper uses a novel type of ordinal analysis, expanding that for $\mathrm{KP}$ to the higher set-theoretic universe while at the same time treating the world of subsets of $N$ as an unanalyzed class-sized urelement structure.

Among the systems of reverse mathematics, for which we determine the admissible extension, are ${\Pi }_1^1\text{-}{\mathrm{CA}}_0$ and ${\mathrm{ATR}}_0$ as well as the theory of bar induction, $\mathrm{BI}$.