A model is said to be -like if but for all , . In this paper, we shall study sentences true in -like models of arithmetic, especially in the cases when is singular. In particular, we identify axiom schemes true in such models which are particularly `natural' from a combinatorial or model-theoretic point of view and investigate the properties of models of these schemes.

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 κ-like model where κ is an inaccessible cardinal. Such T proves the Inaccessibility Scheme. In this paper we study elementary end extensions of models of the inaccessibility scheme.