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Miroslav Repick´y A proof of the independence of the Axiom of Choice from the Boolean Prime Ideal Theorem

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Miroslav Repick´ y

A proof of the independence of the Axiom of Choice from the Boolean Prime Ideal Theorem

Comment.Math.Univ.Carolin. 56,4 (2015) 543 –546.

Abstract: 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-L¨ auchli partition theorem and instead we reduce the proof to its elementary case.

Keywords: Boolean Prime Ideal Theorem; the Axiom of Choice

AMS Subject Classification: Primary 03E35, Secondary 03E25, 03E40, 03E45 References

[1] Halpern J.D., L¨auchli H.,A partition theorem, Trans. Amer. Math. Soc.124(1966), 360–367.

[2] Halpern J.D., L´evy 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.

[3] Jech T.,Set Theory, Academic Press, New York-London, 1978.

[4] Jech T.,Set Theory, the third millennium edition, revised and expanded, Springer Monographs in Mathematics, Springer, Berlin, 2003.

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