# Tiered Arithmetics

• Helmut Schwichtenberg
• Stanley S. Wainer
Chapter
Part of the Outstanding Contributions to Logic book series (OCTR, volume 13)

## Abstract

In his paper “Logics for Termination and Correctness of Functional Programs, II. Logics of Strength PRA” [5] Feferman was concerned with the problem of how to guarantee the feasibility (or at least the subrecursive complexity) of functions definable in certain logical systems. His ideas have influenced much subsequent work, for instance the final chapter of [13]. There, linear two-sorted systems $$\mathrm {LT({;})}$$ (a version of Gödel’s T) and $$\mathrm {LA({;})}$$ (a corresponding arithmetical theory) of polynomial-time strength were introduced. Here we extend $$\mathrm {LT({;})}$$ and $$\mathrm {LA({;})}$$ in such a way that some forms of non-linearity are covered as well. This is important when one wants to deal on the proof level with particular algorithms, not only with the functions they compute. Examples are divide-and-conquer approaches as in treesort, and the first of two main sections here gives a detailed analysis of this. The second topic treated heads in a quite different direction, though again its roots lie in the final chapter of [13]. Instead of just two sorts we consider transfinite ramified sequences of them, or “tiers”; ordinally labelled copies of the natural numbers, respecting certain pointwise orderings. A hierarchy of infinitary arithmetical theories $$\mathrm {EA}(I_\alpha )$$ is devised, $$I_\alpha$$ designating the top tier. These are weak numerical analogues of the iterated inductive definitions underpinning much of Feferman’s fundamental work over decades; see for example his survey [6] and the technical classic [3]. The computational strength of $$\mathrm {EA}(I_\alpha )$$ is summarized thus: it proves the totality of all functions elementary in the Fast-Growing $$F_\alpha$$. A “pointwise” concept of transfinite induction then provides an ordinal measure of strength, but this is a weak, finitistic analogue of the usual notions, related to the Slow-Growing hierarchy.

## Keywords

Polynomial time Linear two-sorted arithmetic Program extraction Tiered arithmetic Fast and slow-growing hierarchies Pointwise transfinite induction.

## 2010 Mathematics Subject Classification

03D20 03D15 03F05

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