Admissible extensions of subtheories of second order arithmetic

TL;DR

This paper investigates how certain extensions of second order arithmetic theories can incorporate both the axioms of the theories and a set-theoretic hierarchy based on Kripke-Platek set theory, exploring their structural properties.

Contribution

It introduces a framework for admissible extensions of reverse mathematics theories that combine second order arithmetic with set-theoretic hierarchies.

Findings

Characterization of admissible extensions of theories T

Analysis of the structural properties of the combined models

Insights into the interaction between reverse mathematics and set theory

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 N 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, KP.