A note on the deductive strength of the Nielsen‐Schreier theorem
Eleftherios Tachtsis · Mathematical logic quarterly · 2018
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”.