Boolean Prime Ideal Theorem; the Axiom of Choice
We present a proof of the Boolean Prime Ideal Theorem in a transitive model of ZF in which the Axiom of Choice does not hold. We omit the argument based on the full Halpern-Läuchli partition theorem and instead we reduce the proof to its elementary case.
 Halpern J.D., Lévy A.: The Boolean Prime Ideal Theorem does not imply the Axiom of Choice
. In: Axiomatic Set Theory, Proceedings of Symposia in Pure Mathematics, vol. XIII, Part I, pp. 83–134, AMS, Providence, 1971. MR 0284328
| Zbl 0233.02024
 Jech T.: Set Theory
. the third millennium edition, revised and expanded, Springer Monographs in Mathematics, Springer, Berlin, 2003. MR 1940513
| Zbl 1007.03002