Fragments of Arithmetic and true sentences

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.Junta de Andalucía TIC-13

The complexity of classification problems for models of arithmetic

We observe that the classification problem for countable models of arithmetic is Borel complete. On the other hand, the classification problems for finitely generated models of arithmetic and for recursively saturated models of arithmetic are Borel; we investigate the precise complexity of each of these. Finally, we show that the classification problem for pairs of recursively saturated models and for automorphisms of a fixed recursively saturated model are Borel complete.Comment: 15 page