Provable wellorderings of formal theories for transfinitely iterated inductive definitions

By [12] we know that transfinite induction up to Θ eΩ N +1 0 is not provable in ID N , the theory of N -times iterated inductive definitions. In this paper we will show that conversely transfinite induction up to any ordinal less than Θ eΩ N +1 0 is provable in ID N i , the intuitionistic version of ID N , and extend this result to theories for transfinitely iterated inductive definitions. In [14] Schutte proves the wellordering of his notational systems using predicates is wellordered) with M κ ≔ { x ∈ and 0 ≤ κ ≤ N . Obviously the predicates are definable in ID N i with the defining axioms: where Prog [ M κ , X ] means that X is progressive with respect to M κ , i.e. The crucial point in Schutte's wellordering proof is Lemma 19 [14, p. 130] which can be modified to where TI[ M κ + 1 , a ] is the scheme of transfinite induction over M κ + 1 up to a .