Quantifier-free induction schema and the least element principle

Abstract

We consider the quantifier-free induction schema and the least element principle in the language of elementary arithmetic enriched by a free function symbol ƒ. Some stronger, iterated versions of these schemata are also considered. We show that the iterated induction schema does not prove the existence of the maximum of ƒ on every finite interval. A similar result obtained for the non-iterated least element principle. At the same time, already the doubly iterated least element principle for quantifier-free formulas proves the existence of the maximum of ƒ. We also obtain additional results on the relationships between the two schemata, and outline the connections with the induction schema and the least element principle for decidable relations.