Equivalence of the Induction Schema and the Least Number Principle for Open Formulas (original) (raw)

Let LA be the usual language for arithmetic. Let ϕ(x) be an LAformula. ϕ(x) may contain free variables distinct from x as parameters. We consider the following two schemata. (I ϕ(x)) ϕ(0) ∧ ∀x(ϕ(x) → ϕ(x + 1)) → ∀xϕ(x), (L ϕ(x)) ∃xϕ(x) → ∃x(∀y < x ¬ϕ(y) ∧ ϕ(x)). They are called the induction schema and the least number principle, respectively. IOpen, LOpen will denote the theory P A − ∪ {I ϕ(x) | ϕ(x) : open}, P A − ∪ {L ϕ(x) | ϕ(x) : open}, respectively. In this paper we prove the equivalence of IOpen and LOpen. Van den Dries [v.d.D] noted that this can be proven model theoretically by using ideas in the proof of Shepherdson's theorem in [S1]. Our proof is syntactical and not model theoretical.