A note on the deductive strength of the Nielsen‐Schreier theorem |
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| Abstract: | 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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