I have become aware of an amazing phenomenon from a myriad of questions and answers here on MathOverflow: many of the results that I would typically prove using the Axiom of Choice can actually be proved using the logically weaker Boolean Prime Ideal Theorem. Apparently, the following can be proved using the BPI theorem:

- Every nonzero commutative unital ring contains a prime ideal
- Every field has an algebraic closure
- The Hahn-Banach Theorem
- Every formally real field can be totally ordered

I tempted to believe that almost every application of the Axiom of Choice that I would use in my everyday work actually follows from the Boolean Prime Ideal Theorem (except for those obviously equivalent to the Axiom of Choice, such as: every epimorphism in the category of sets is split).

**Thus I wish to ask:** Given that I know how to prove many "everyday" mathematical statements using the Axiom of Choice, how can I learn to prove them with the weaker Boolean Prime Ideal Theorem?

To make the question more precise, let's suppose that I know how to prove a result using Zorn's Lemma. Is there some standard method that I can use to put in just a bit more work and extract the same theorem using the BPI theorem?

The best I can think to do would be to use the restatement of the BPI theorem as the existence of ultrafilters. I have only ever seen this stated as being equivalent to the existence of an ultrafilter on every set. So first I should probably ask: does the BPI theorem that every filter on a ("nice enough") poset is contained in an ultrafilter?

If I have a poset $\mathcal{P}$ satisfying the usual Zorn property, the nice thing about Zorn's lemma is that it gives me an actual maximal element of the poset $\mathcal{P}$. The problem I see with existence of ultrafilters in $\mathcal{P}$ is that I do not obtain an actual element of $\mathcal{P}$, but a (rather large) subset. Is there a standard way that I can translate this back into an element of my poset $\mathcal{P}$? (For instance, should I pass to a least upper bound, because the posets that usually arise in Zorn applications are upper-complete?)

For instance, let's consider the first of the examples on the list above. Say $R$ is a commutative unital ring, and let $I \subsetneq R$ be a proper ideal of $R$. How can I prove using the BPI theorem that there is a prime ideal of $R$ containing $I$? I would typically consider the poset $\mathcal{P}$ of proper ideals of $R$ containing $I$, use Zorn's lemma to produce a maximal element $M$ of $\mathcal{P}$ (which will actually be a maximal ideal), and then prove that $M$ is prime. Supposing that the BPI theorem really gives me existence of ultrafilters in $\mathcal{P}$, I would probably start by considering the filter $\mathcal{F}$ of ideals containing $I$, and then pass to an ultrafilter containing this. Is there some way for me to obtain a prime ideal from such an ultrafilter? (And as I asked above, do I really obtain such an ultrafilter from the BPI theorem?)

I have included the "reference-request" tag because I would be satisfied with a reference to a survey paper that would teach me how to effectively apply the BPI theorem in a number of instances (including the ones above). While I would certainly be interested in references that show how to prove the above results (and any others!) using the BPI theorem, this would not be the kind of answer that I seek.

oftenvery different from one another. They might appeal to various different equivalents of $\sf BPI$ (e.g. the compactness theorem of first-order logic; or the Tychonoff theorem for Hausdorff spaces). $\endgroup$ – Asaf Karagila Apr 9 '15 at 16:20