Let n be a positive integer. By a βn-model we mean an ω-model which is elementary with respect to Σ1n formulas. We prove the following βn-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 βn-model of S, then there exists a βn-model of S + “there is no countable βn-model of S”. We also prove a βn-model version of Löb’s Theorem. As a corollary, we obtain a βn-model which is not a βn+1-model.
"An incompleteness theorem for βn-models." J. Symbolic Logic 69 (2) 612 - 616, June 2004. https://doi.org/10.2178/jsl/1082418545