Constructing κ‐like Models of Arithmetic

A model ( M , <, …) is κ‐like if M has cardinality κ but, for all α ∈ M , the cardinality of { x ∈ 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 M ⊧ Π 2 −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 M ⊧ B ∑ n +exp+¬ I ∑ n there is a κ‐like model K elementarily equivalent to M .