A flexible type system for the small Veblen ordinal

Abstract

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.