Peano Corto and Peano Basso: A Study of Local Induction in the Context of Weak Theories
Article first published online: 8 FEB 2014
© 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
Mathematical Logic Quarterly
Volume 60, Issue 1-2, pages 92–117, February 2014
How to Cite
Visser, A. (2014), Peano Corto and Peano Basso: A Study of Local Induction in the Context of Weak Theories. Mathematical Logic Quarterly, 60: 92–117. doi: 10.1002/malq.201200102
- Issue published online: 8 FEB 2014
- Article first published online: 8 FEB 2014
- Manuscript Accepted: 7 JAN 2014
- Manuscript Revised: 26 SEP 2013
- Manuscript Received: 30 NOV 2012
In this paper we study local induction w.r.t. Σ1-formulas over the weak arithmetic . The local induction scheme, which was introduced in , says roughly this: for any virtual class that is progressive, i.e., is closed under zero and successor, and for any non-empty virtual class that is definable by a Σ1-formula without parameters, the intersection of and is non-empty. In other words, we have, for all Σ1-sentences S, that S implies , whenever 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 . 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, and . On the one hand, Peano Corto and Peano Basso are very weak: they are locally cut-interpretable in . 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 , the theory of parameter-free Π1-induction.