An ordinal analysis of stability

Abstract

This paper is the first in a series of three which culminates in an ordinal analysis of Π12-comprehension. On the set-theoretic side Π12-comprehension corresponds to Kripke-Platek set theory, KP, plus ∑1-separation. The strength of the latter theory is encapsulated in the fact that it proves the existence of ordinals π such that, for all β>π, π is β-stable, i.e. L π is a ∑1-elementary substructure of L β . The objective of this paper is to give an ordinal analysis of a scenario of not too complicated stability relations as experience has shown that the understanding of the ordinal analysis of Π12-comprehension is greatly facilitated by explicating certain simpler cases first. This paper introduces an ordinal representation system based on ν-indescribable cardinals which is then employed for determining an upper bound for the proof-theoretic strength of the theory KPi+ ∀ρ ∃π π is π+ρ-stable, where KPi is KP augmented by the axiom saying that every set is contained in an admissible set. © Springer-Verlag 2004.