An incompleteness theorem for βn-models

Abstract

Let n be a positive integer. By a β_n-model we mean an ω-model which is elementary with respect to Σ¹_n formulas. We prove the following β_n-model version of Gödel’s Second Incompleteness Theorem. For any recursively axiomatized theory S in the language of second order arithmetic, if there exists a β_n-model of S, then there exists a β_n-model of S + “there is no countable β_n-model of S”. We also prove a β_n-model version of Löb’s Theorem. As a corollary, we obtain a β_n-model which is not a β_n+1-model.