Let $X$ be a set, $x',y'$ elements of $X$ and let $P$ be the set of all partial orderings of $X$; is it true that there exists $\leq \in P$ such that either $x\leq y$ or $y\leq x$ (or both)? i.e. is is true that there must be some partial ordering of $X$ such that $x$ and $y$ can be compared? Why is that so?
(Note: I'm aware of the fact that if we assume the well-ordering principle the answer is yes, I'm wondering if the same is true if we do not assume it)
Best regards,
lorenzo.