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Keywords:
Boolean Prime Ideal Theorem; the Axiom of Choice
Boolean Prime Ideal Theorem; the Axiom of Choice
Summary:
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.
References:
[1] Halpern J.D., Läuchli H.: A partition theorem. Trans. Amer. Math. Soc. 124 (1966), 360–367. DOI 10.1090/S0002-9947-1966-0200172-2 | MR 0200172 | Zbl 0158.26902
[2] 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
[3] Jech T.: Set Theory. Academic Press, New York-London, 1978. MR 0506523 | Zbl 1007.03002
[4] Jech T.: Set Theory. the third millennium edition, revised and expanded, Springer Monographs in Mathematics, Springer, Berlin, 2003. MR 1940513 | Zbl 1007.03002
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