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

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 $$x (namely: $$x\lesssim y$$ but not $$y\lesssim 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 $$x\sim y$$ iff $$x\lesssim y$$ and $$y\lesssim x$$, 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).

• I have to check more carefully, but I am pretty sure that D is badly argued: it's not clear that $\preceq$ as I defined it is transitive. Sep 4 '20 at 14:16