The Theory of $\kappa$-like Models of Arithmetic

Abstract

A model $(M,<,\ldots)$ is said to be $\kappa$-like if $\mbox{card $\;$} M =\kappa$ but for all $a\in M$, $\mbox{card}\{x \in M: x< a\}< \kappa$. In this paper, we shall study sentences true in $\kappa$-like models of arithmetic, especially in the cases when $\kappa$ 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.