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

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.

Definably extending partial orders in totally ordered structures

“…We note the following, which was observed by Felgner and Truss [, Lemma ]. Corollary If M is well‐ordered, then M extends partial orders. Proof We show that M satisfies the hypothesis of Theorem .…”

“…In particular, our results imply that both weakly o‐minimal structures (hence all o‐minimal structures) and quasi‐o‐minimal structures extend partial orders (Corollaries and ). Our results also imply that well‐ordered structures extend partial orders (Corollary ), although this was already known from . In the cases we consider, the definable linear order extending a given definable partial order is uniformly definable from the partial order.…”

“…Indeed, the Order Extension Principle, a set‐theoretic axiom which turns Szpilrajn's Theorem into an axiom, may be viewed as a weak choice‐like axiom (cf. for a discussion). Generally speaking, such proofs do not do not give very much information about a linear order that extends the given partial order.…”

On a Possible Continuous Analogue of the Szpilrajn Theorem and its Strengthening by Dushnik and Miller

“…In the context of algebraic structures we want to mention the papers by Kontolatou and Stabakis [36] and Martínez-Legaz and Singer [39] and the more recent paper by Fishburn [25] where possible generalizations of the Szpilrajn theorem to additive partial orders are discussed. Felgner and Truss [22] consider the Szpilrajn property or order-extension property, which means that every partial order can be extended or refined to a total order in order to discuss the question if the Szpilrajn property implies the prime ideal theorem which states that every Boolean algebra has a prime ideal. In Petri's concurrency theory one usually works with a discrete version of Dushnik's and Miller's strengthening of the Szpilrajn theorem.…”