Archive for Mathematical Logic

, Volume 44, Issue 1, pp 1-62

First online:

An ordinal analysis of stability

  • Michael RathjenAffiliated withDepartment of Pure Mathematics, University of Leeds Email author 

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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. L π is a Σ1-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+ ∀ρπ π is π+ρ-stable, where KPi is KP augmented by the axiom saying that every set is contained in an admissible set.