Is $\textsf{BPIT}$ equivalent to some ordering principle?

Question

Working in $\mathsf{ZF}$, is $\mathsf{BPIT}$ (Boolean Prime Ideal Theorem) equivalent to some statement of the form "every set can be ***ly ordered"? I know that $\mathsf{BPIT}$ implies that every set can be linearly ordered, but it does not imply that every set can be well-ordered (because $\mathsf{BPIT}$ is strictly weaker than $\mathsf{AC}$).

Answer 1

The answer is no.

First of all, $\sf BPIT$ is strictly stronger than "every set can be linearly ordered", it is even stronger than "every partial order can be extended to a linear order".

Secondly, in the paper

David Pincus, The dense linear ordering principle, J. Symbolic Logic 62 (1997), no. 2, 438--456. MR 1464107

the author proves that the following implications are irreversible, even if we assume $\sf BPIT$:

1. Every set can be well-ordered.
2. Every set can be injectively mapped into a power set of an ordinal (which is densely ordered by the lexicographic ordering).
3. Every set can be densely ordered.
4. Every set can be linearly ordered.

It follows, if so, that $\sf BPIT$ is too strong and too weak to be equivalent to any reasonable "Every set can be ***ly ordered".