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Pointwise Transfinite Induction and a Miniaturized Predicativity

  • Stanley S. WainerEmail author
Chapter
Part of the Progress in Computer Science and Applied Logic book series (PCS, volume 28)

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.

Keywords

Computable Function Infinitary System Computation Rule Peano Arithmetic Fundamental Sequence 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.University of LeedsLeedsUK

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