A note on the deductive strength of the Nielsen‐Schreier theorem

Mathematical Logic Quarterly 64 (3):173-177 (2018)
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We show that the Boolean Prime Ideal Theorem () does not imply the Nielsen‐Schreier Theorem () in, thus strengthening the result of Kleppmann from “Nielsen‐Schreier and the Axiom of Choice” that the (strictly weaker than ) Ordering Principle () does not imply in. We also show that is false in Mostowski's Linearly Ordered Model of. The above two results also settle the corresponding open problems from Howard and Rubin's “Consequences of the Axiom of Choice”.



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Subgroups of a free group and the axiom of choice.Paul E. Howard - 1985 - Journal of Symbolic Logic 50 (2):458-467.
Nielsen‐Schreier and the Axiom of Choice.Philipp Kleppmann - 2015 - Mathematical Logic Quarterly 61 (6):458-465.
Adding dependent choice.David Pincus - 1977 - Annals of Mathematical Logic 11 (1):105.

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