# Strength of claims about extensions of partial preorders and orders to linear ones

MathOverflow Asked by Alexander Pruss on September 6, 2020

Consider these two axioms:

1. Every partial order extends to a linear order.
2. Every partial preorder (reflexive and transitive relation) extends to a linear preorder while preserving strict orderings: i.e., whenever $$x (namely: $$xlesssim y$$ but not $$ylesssim x$$) holds in the original order, it holds in the extended order.

Question: Does 2 imply 1 in ZF?

Notes:

A. In ZF, Boolean Prime Ideal implies 1 (but not conversely if ZF is consistent), and 1 implies 2 (take the quotient of the preordered set under $$sim$$, where $$xsim y$$ iff $$xlesssim y$$ and $$ylesssim x$$, and apply 1).

B. Also, 2+(every set has a linear order) implies 1. (Use 2 to extend the partial order $$le$$ to a total preorder $$lesssim$$ preserving strict orderings. We now need to turn $$lesssim$$ into a linear order. To do that, we just need to linearly order within each equivalence class under $$sim$$. Do that by taking a linear order on the whole set and using that to induce the order in each equivalence class—though not between them.)

C. Claim 2 implies Banach-Tarski and thus the existence of nonmeasurable sets (Pawlikowski’s proof of Banach-Tarski from Hahn-Banach can be adapted), and hence 2 is not provable in ZF (if ZF is consistent).