Peano Corto and Peano Basso: A Study of local induction in the context of weak theories

Abstract

In this paper we study local induction w.r.t. Σ1-formulas over the weak arithmetic PA- The local induction scheme, which was introduced in, says roughly this: for any virtual class X that is progressive, i.e., is closed under zero and successor, and for any non-empty virtual class Y that is definable by a Σ1-formula without parameters, the intersection of X and Y is non-empty. In other words, we have, for all Σ1-sentences S, that S implies SX, whenever X is progressive. Since, in the weak context, we have (at least) two definitions of Σ1, we obtain two minimal theories of local induction w.r.t. Σ1-formulas, which we call Peano Corto and Peano Basso. In the paper we give careful definitions of Peano Corto and Peano Basso. We establish their naturalness both by giving a model theoretic characterization and by providing an equivalent formulation in terms of a sentential reflection scheme. The theories Peano Corto and Peano Basso occupy a salient place among the sequential theories on the boundary between weak and strong theories. They bring together a powerful collection of principles that is locally interpretable in PA- Moreover, they have an important role as examples of various phenomena in the metamathematics of arithmetical (and, more generally, sequential) theories. We illustrate this by studying their behavior w.r.t. interpretability, model interpretability and local interpretability. In many ways the theories are more like Peano arithmetic or Zermelo Fraenkel set theory, than like finitely axiomatized theories as Elementary Arithmetic, |Σ1 and ACA0. On the one hand, Peano Corto and Peano Basso are very weak: they are locally cut-interpretable in PA- On the other hand, they behave as if they were strong: they are not contained in any consistent finitely axiomatized arithmetical theory, however strong. Moreover, they extend IΠ1-, the theory of parameter-free Π1-induction.