Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-28T08:02:39.622Z Has data issue: false hasContentIssue false

CONSTRUCTING κ-LIKE MODELS OF ARITHMETIC

Published online by Cambridge University Press:  01 February 1997

RICHARD KAYE
Affiliation:
School of Mathematics and Statistics, The University of Birmingham, Edgbaston, Birmingham B15 2TT, UK. E-mail: [email protected]
Get access

Abstract

A model (M, <, …) is κ-like if M has cardinality κ but, for all α ∈ M, the cardinality of {xM [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 Πn−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.

Type
Research Article
Copyright
The London Mathematical Society 1997

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)