Model Theory of the Inaccessibility Scheme

Suppose L = {<, . . .} is any countable first order language in which < is interpreted as a linear order. Let T be any complete first order theory in the language L such that T has a kappa-like model where kappa is an inaccessible cardinal. Such T satisfies the Inaccessibility Scheme. In this paper we study model theory of the inaccessibility scheme at the level of the existence of elementary end extensions for various models of it.