Pointwise Transfinite Induction and a Miniaturized Predicativity

Abstract

The basis of this work is Leivant’s [6] theory of ramified induction over N, which has elementary recursive strength. It has been redeveloped and extended in various ways by many people; for example, Spoors and Wainer [13] built a hierarchy of ramified theories whose strengths correspond to the levels of the Grzegorczyk hierarchy. Here, a further extension of this hierarchy is developed, in terms of a predicatively generated infinitary calculus with stratifications of numerical inputs up to and including level $\omega$. The autonomous ordinals are those below $\Gamma _0$, but they are generated according to a weak (though quite natural) notion of transfinite induction whose computational strength is “slow” rather than “fast” growing. It turns out that the provably computable functions are now those elementary recursive in the Ackermann function (i.e. Grzegorczyk’s $\omega$th level). All this is closely analogous to recent works of Jäger and Probst [5] and Ranzi and Strahm [9] on iterated stratified inductive definitions, but their theories have full, complete induction as basis, whereas ours have only a weak, ramified form of numerical induction at bottom.

For Gerhard Jäger at his 60th.