Archive for Mathematical Logic 58 (5-6):711-751 (2019)

Abstract
We introduce and analyze two theories for typed inductive definitions and establish their proof-theoretic ordinal to be the small Veblen ordinal \. We investigate on the one hand the applicative theory \ of functions, inductive definitions, and types. It includes a simple type structure and is a natural generalization of S. Feferman’s system \\). On the other hand, we investigate the arithmetical theory \ of typed inductive definitions, a natural subsystem of \, and carry out a wellordering proof within \ that makes use of fundamental sequences for ordinal notations in an ordinal notation system based on the finitary Veblen functions. The essential properties for describing the ordinal notation system are worked out.
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DOI 10.1007/s00153-019-00658-x
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Proof-Theoretic Investigations on Kruskal's Theorem.Michael Rathjen & Andreas Weiermann - 1993 - Annals of Pure and Applied Logic 60 (1):49-88.
Reflections on Reflections in Explicit Mathematics.Gerhard Jäger & Thomas Strahm - 2005 - Annals of Pure and Applied Logic 136 (1-2):116-133.

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