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

U. Felgner and J. K. Truss
The Journal of Symbolic Logic
Vol. 64, No. 1 (Mar., 1999), pp. 199-215
DOI: 10.2307/2586759
Stable URL: http://www.jstor.org/stable/2586759
Page Count: 17
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The Independence of the Prime Ideal Theorem from the Order-Extension Principle
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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.

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