Fragments of Arithmetic and true sentences

Abstract

By a theorem of R. Kaye, J. Paris and C. Dimitracopoulos, the class of the ¦n+1–sentences true in the standard model is the only (up to deductive equivalence) consistent ¦n+1–theory which extends the scheme of induction for parameter free ¦n+1–formulas. Motivated by this result, we present a systematic study of extensions of bounded quantifier complexity of fragments of first–order Peano Arithmetic. Here, we improve that result and show that this property describes a general phenomenon valid for parameter free schemes. As a consequence, we obtain results on the quantifier complexity, (non)finite axiomatizability and relative strength of schemes for ¢n+1–formulas.