Completeness theorems, incompleteness theorems and models of arithmetic

Abstract

Let A \mathcal {A} be a consistent extension of Peano arithmetic and let A n 0 \mathcal {A}_n^0 denote the set of Π n 0 \Pi _n^0 consequences of A \mathcal {A} . Employing incompleteness theorems to generate independent formulas and completeness theorems to construct models, we build nonstandard models of A n + 2 0 \mathcal {A}_{n + 2}^0 in which the standard integers are Δ n + 1 0 \Delta _{n + 1}^0 -definable. We thus pinpoint induction axioms which are not provable in A n + 2 0 \mathcal {A}_{n + 2}^0 ; in particular, we show that (parameter free) Δ 1 0 \Delta _1^0 -induction is not provable in Primitive Recursive Arithmetic. Also, we give a solution of a problem of Gaifman on the existence of roots of diophantine equations in end extensions and answer questions about existentially complete models of A 2 0 \mathcal {A}_2^0 . Furthermore, it is shown that the proof of the Gödel Completeness Theorem cannot be formalized in A 2 0 \mathcal {A}_2^0 and that the MacDowell-Specker Theorem fails for all truncated theories A n 0 \mathcal {A}_n^0 .