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The Existence of Level Sets in a Free Group Implies the Axiom of Choice
Author(s) -
Howard Paul E.
Publication year - 1987
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.19870330406
Subject(s) - axiom of choice , axiom , citation , group (periodic table) , mathematical economics , mathematics , combinatorics , computer science , library science , set theory , programming language , set (abstract data type) , chemistry , geometry , organic chemistry
Since all known proofs of the NIELSEN-SCHREIER theorem (every subgroup of a free group is free, henceforth denoted by (NS)) make use of the axiom of choice (AC) it is reasonable to conjecture that some form of (AC) is necessary to prove (NS). (See [ l ] , [3], [5] and [S] for several proofs of (NS).) This conjecture was verified by LAUCHLI in [4], where it was shown that the negation of (NS) is consistent with ZFA, ZermeloFraenkel set theory weakened to permit the existence of atoms. The result was strengthened in [2], where it was shown that (NS) implies the axiom of choice for sets of finite sets.