Constructing Κ-like Models of Arithmetic

A model (M,!,...) is κ-like if M has cardinality κ but, for all a `M, the cardinality of 2x `M : x! a ́ is strictly less than κ. In this paper we shall give constructions of κ-like models of arithmetic satisfying an arbitrarily large finite part of PA but not PA itself, for various singular cardinals κ. The main results are : (1) for each countable nonstandard MzΠ # ®Th(PA) with arbitrarily large initial segments satisfying PA and each uncountable κ of cofinality ω there is a cofinal extension K of M which is κ-like ; also hierarchical variants of this result for Π n ®Th(PA); and (2) for every n& 1, every singular κ and every MzBΣ n ­exp­| IΣ n there is a κ-like model K elementarily equivalent to M.