Boolean prime ideal theorem

Remarks

In ZFC the proof applies Zorn's lemma to the ideals that contain $I$ and omit $1$. A maximal one $P$ is prime: if $a \notin P$ and $\neg a \notin P$, then the ideals generated by $P \cup \set{a}$ and by $P \cup \set{\neg a}$ both contain $1$, and this forces $1 \in P$. The theorem is usually abbreviated BPI. By duality, BPI is equivalent to the ultrafilter lemma: each filter on a set extends to an ultrafilter. Over ZF, BPI follows from the axiom of choice but does not imply it. Over ZF, BPI is equivalent to the compactness theorem of first-order logic and to Stone's representation theorem for Boolean algebras. Example: in the power set Boolean algebra of a set $X$, with $x \in X$, the family of subsets of $X$ that do not contain $x$ is a prime ideal.

Going deeper

For a student

BPI is the choice principle behind many compactness arguments. It gives the ultrafilter lemma, the compactness theorem of first-order logic, and the representation of every Boolean algebra as an algebra of clopen sets of a Stone space. It also gives Tychonoff's theorem for products of compact Hausdorff spaces, though not for arbitrary compact spaces, which needs full choice. So it sits strictly between ZF and ZFC, and it is a common test of how much choice a theorem really needs.