On the Existence of end Extensions of Models of Bounded Induction

Abstract

This chapter focuses on the existence of end extensions of models of bounded induction. From the results of Friedman, it follows that any nonstandard countable model of ∑ n induction (I∑ n ) for n > 1 is isomorphic to a proper initial segment of itself, and hence, has a proper end extension to a model of I∑ n. This was later extended to the case n = 1. For n = 0, this result is false because in proposition 1, if M and K are models of IΔ 0 , and K is a proper end extension of M (M ⊂ K), then M must also satisfy ∑ 1 collection (B∑ 1 ). However, there are models of IΔ 0 that do not satisfy B∑ 1 and, hence, do not have proper end extensions to models of IΔ 0 . This then raises the question of finding necessary and sufficient conditions on a countable model M of IΔ 0 for M to have a proper end extension to a model of IΔ 0.