Archive for Mathematical Logic

, Volume 44, Issue 1, pp 1–62

An ordinal analysis of stability


    • Department of Pure MathematicsUniversity of Leeds

DOI: 10.1007/s00153-004-0226-2

Cite this article as:
Rathjen, M. Arch. Math. Logic (2005) 44: 1. doi:10.1007/s00153-004-0226-2


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 Σ1-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. Lπ is a Σ1-elementary substructure of Lβ. 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+ ∀ρππ is π+ρ-stable, where KPi is KP augmented by the axiom saying that every set is contained in an admissible set.

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© Springer-Verlag Berlin Heidelberg 2004