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Nielsen‐Schreier and the Axiom of Choice
Author(s) -
Kleppmann Philipp
Publication year - 2015
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.201400046
Subject(s) - axiom of choice , mathematics , zermelo–fraenkel set theory , axiom , axiom independence , urelement , section (typography) , choice function , constructive set theory , mathematical economics , pure mathematics , set (abstract data type) , set theory , computer science , geometry , programming language , operating system
The Nielsen‐Schreier theorem asserts that subgroups of free groups are free. In the first section we show that this theorem does not follow from the Linear Ordering Principle, thus strengthening the fact that it implies the Axiom of Choice for families of finite sets. In the second section, we show that a stronger variant of the Nielsen‐Schreier theorem implies the Axiom of Choice.