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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]
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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

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