A note on the deductive strength of the Nielsen‐Schreier theorem. Issue 3 (3rd July 2018)

• Record Type: Journal Article
• Authors: Tachtsis, Eleftherios
• Is Part Of: Mathematical logic quarterly. Volume 64:Issue 3(2018)
• Page Start: 173
• Page End: 177
• Publication Date: 2018-07-03
• Subjects: Mathematics -- Periodicals; Logic, Symbolic and mathematical -- Periodicals; 511.3
• DOI: 10.1002/malq.201700022
• Languages: English
• ISSNs: 0942-5616

Abstract

Abstract: We show that the Boolean Prime Ideal Theorem ( BPI ) does not imply the Nielsen‐Schreier Theorem ( NS ) in ZF, thus strengthening the result of Kleppmann from "Nielsen‐Schreier and the Axiom of Choice" that the (strictly weaker than BPI ) Ordering Principle ( OP ) does not imply NS in ZF . We also show that NS is false in Mostowski's Linearly Ordered Model of ZFA + BPI . The above two results also settle the corresponding open problems from Howard and Rubin's "Consequences of the Axiom of Choice".