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The independence of the Prime Ideal Theorem from the Order-Extension Principle

Published online by Cambridge University Press:  12 March 2014

U. Felgner  and
J. K. Truss
U. Felgner
Affiliation:
Mathematisches Institut der Universität, Auf Der Morgenstelle 10, 7400 Tübingen, Germany, E-mail: [email protected]
J. K. Truss
Affiliation:
Department of Pure Mathematics, University of Leeds, Leeds LS2 9JT, England, E-mail: [email protected]
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Abstract

It is shown that the boolean prime ideal theorem BPIT: every boolean algebra has a prime ideal, does not follow from the order-extension principle OE: every partial ordering can be extended to a linear ordering. The proof uses a Fraenkel–Mostowski model, where the family of atoms is indexed by a countable universal-homogeneous boolean algebra whose boolean partial ordering has a ‘generic’ extension to a linear ordering. To illustrate the technique for proving that the order-extension principle holds in the model we also study Mostowski's ordered model, and give a direct verification of OE there. The key technical point needed to verify OE in each case is the existence of a support structure.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 64 , Issue 1 , March 1999 , pp. 199 - 215
Copyright
Copyright © Association for Symbolic Logic 1999

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