CONSTRUCTING κ-LIKE MODELS OF ARITHMETIC

Abstract

A model (M, <, …) is κ-like if M has cardinality κ but, for all α ∈ M, the cardinality of {x ∈ M [ratio ] 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 [models ] Π₂−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 [ges ] 1, every singular κ and every M [models ] B[sum ]_n+ exp+¬ I[sum ]_n there is a κ-like model K elementarily equivalent to M.