A Type Theory for Iterated Inductive Definitions

We introduce a type theory FAω, which has at least the strength of finitely iterated inductive definitions ID<ω. This type theory has as ground types trees with finitely many branching degrees (so called free algebras). We introduce an equality in this theory, without the need for undecidable prime formulas. Then we give a direct well-ordering proof for this theory by representing a ordinal denotation system in the iteration of Kleene’s O. This can be easily done, by introducing functions on the trees, which correspond to the functions in the ordinal denotation system. The proof shows, that FAω proofs transfinite induction up to D0Dn0, which shows, that the strength of FAω is at least ID<ω. It seems to be obvious, that this bound is sharp. 1 Definition of the type theory FAω Definition 1.1 The type theory FAω is defined as follows: (a) The ground types are defined inductively by: If n ≥ 0 and α1, . . . , αn are ground types, then (α1, . . . , αn) is a ground type. The type (α1, . . . , αn) should be the type of well-founded trees with branching degrees α1, . . . , αn. (b) Ground types are types, and if α, β are types then (α → β) is a type. We will omit brackets, using the usual conventions. (c) If α = (α1, . . . , αn) is a ground type, then for i = 1, . . . , n C α i is a constant of type (αi → α) → α (C α i are the constructors for this type) and if α is as before and σ a type, then we have the recursion constant Rα,σ of type ((α1 → α) → (α1 → σ) → σ) → · · · → ((αn → α) → (αn → σ) → σ) → α → σ We will write (C1 : α1, . . . , Cn : αn) for (α1, . . . , αn) to indicate, that Ci are names for C i .