The Journal of Symbolic Logic

Research Article

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

U. Felgnera1 and J. K. Trussa2

a1 Mathematisches Institut der Universität, Auf Der Morgenstelle 10, 7400 Tübingen, Germany, E-mail:

a2 Department of Pure Mathematics, University of Leeds, Leeds LS2 9JT, England, E-mail:


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.

(Received March 17 1997)

(Revised June 14 1997)