Archive for Mathematical Logic

, Volume 58, Issue 5–6, pp 711–751 | Cite as

A flexible type system for the small Veblen ordinal

  • Florian Ranzi
  • Thomas StrahmEmail author


We introduce and analyze two theories for typed (accessible part) inductive definitions and establish their proof-theoretic ordinal to be the small Veblen ordinal \(\vartheta \Omega ^\omega \). We investigate on the one hand the applicative theory \(\mathsf {FIT}\) of functions, (accessible part) inductive definitions, and types. It includes a simple type structure and is a natural generalization of S. Feferman’s system \(\mathrm {QL}(\mathsf {F}_\mathsf {0}\text {-}\mathsf {IR}_{N})\). On the other hand, we investigate the arithmetical theory \(\mathsf {TID}\) of typed (accessible part) inductive definitions, a natural subsystem of \(\mathsf {ID}_1\), and carry out a wellordering proof within \(\mathsf {TID}\) 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.


Proof theory Inductive definitions Applicative theories Small Veblen ordinal Finitary Veblen functions Metapredicativity Higher types Subsystems of second-order arithmetic 

Mathematics Subject Classification

03F03 03F15 03F35 03F50 


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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institut für InformatikUniversität BernBernSwitzerland

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