An ordinal analysis of stability

This paper is the rst in a series of three which culminates in an ordinal analysis of 1 2-comprehension. On the set-theoretic side 1 2-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 such that, for all > , is-stable, i.e. L is a 1-elementary substructure of L. The objective of this paper is to give an ordinal analysis of not too complicated stability relations as experience has shown that the understanding of the ordinal analysis of 1 2-comprehension is greatly facilated by explicating certain simpler cases rst. This paper introduces an ordinal representation system based on-inaccessible cardinals which is then employed for determining an upper bound for the proof{theoretic strength of the theory KPi + 8 9 is +-stable, where KPi is KP augmented by the axiom saying that every set is contained in an admissible set.