The theory of \(\kappa\)-like models of arithmetic
type: Item
instance of: scholarly article
title: The theory of \(\kappa\)-like models of arithmetic
zbMATH Open document ID: 0848.03019
published in: Notre Dame Journal of Formal Logic
publication date: 1996-10-27T00:00:00Z
review text: A linearly ordered model is \(\kappa\)-like if it is of power \(\kappa\), and each of its proper initial segments is of cardinality smaller than \(\kappa\). In the search for the theory of all \(\kappa\)-like models of PA, the author describes axiom schemes true in all \(\kappa\)-like models of PA, and discusses their model-theoretic properties. The schemes are suitable formalizations of: the statement that in a \(\kappa\)-like model there is no one-to-one map from the model to a proper initial segment; various forms of the pigeonhole principle; collection scheme; an indiscernibility schema related to a general theorem of Keisler on \(\kappa\)-like models for singular strong limits \(\kappa\). The concept of tree indiscernibility related to the work of Paris and Mills on cardinalities of initial segments of models of PA is also discussed. The paper is intended to be read in conjunction with another paper of the author's [``Constructing \(\kappa\)-like models of arithmetic'', J. Lond. Math. Soc. (to appear)].
Mathematics Subject Classification ID: 03C62
Mathematics Subject Classification ID: 03F30
zbMATH DE Number: 881256
zbMATH Keywords: fragments of arithmetic
zbMATH Keywords: indiscernibles
zbMATH Keywords: \(\kappa\)-like models
zbMATH Keywords: pigeonhole principle
zbMATH Keywords: collection scheme
author: Richard Kaye
reviewed by: Roman Kossak
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1305/ndjfl/1040136915
OpenAlex ID: W1532715505
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DOI: 10.1305/NDJFL/1040136915
description: scientific article; zbMATH DE number 881256