Arithmetic transfinite induction and recursive well-orderings

Abstract

A uniform, algebraic proof that every number-theoretic assertion provable in any of the intuitionistic theories T listed below has a well-founded recursive proof tree (demonstraby in T) is given. Thus every such assertion is provable by transfinite induction over some primitive recursive well-ordering. T can be higher order number theory, set theory, or its extensions equiconsistent with large cardinals. It is shown that there is a number-theoretic assertion B( n) (independent of T) with a parameter n such that any primitive recursive linear ordering R on ω for which transfinite induction on R for B( n) is provable in T is in fact a well-ordering.