Fragments of Arithmetic and true sentences
Andrés Cordón–Franco, A. Fernández‐Margarit, Francisco Martín · Mathematical logic quarterly · 2005
By a theorem of R. Kaye, J. Paris and C. Dimitracopoulos, the class of the Πn+1-sentences true in the standard model is the only (up to deductive equivalence) consistent Πn+1-theory which extends the scheme of induction for parameter free Πn+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 Δn+1-formulas. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)