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A note on the deductive strength of the Nielsen‐Schreier theorem
Author(s) -
Tachtsis Eleftherios
Publication year - 2018
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/malq.201700022
Subject(s) - mathematics , axiom of choice , axiom , ideal (ethics) , discrete mathematics , pure mathematics , set theory , law , set (abstract data type) , computer science , geometry , political science , programming language
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”.