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Bases, spanning sets, and the axiom of choice
Author(s) -
Howard Paul
Publication year - 2007
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.200610043
Subject(s) - mathematics , axiom of choice , axiom , assertion , basis (linear algebra) , space (punctuation) , vector space , statement (logic) , element (criminal law) , combinatorics , choice function , set (abstract data type) , discrete mathematics , mathematical economics , pure mathematics , set theory , computer science , epistemology , philosophy , geometry , political science , law , programming language , operating system
Two theorems are proved: First that the statement “there exists a field F such that for every vector space over F , every generating set contains a basis” implies the axiom of choice. This generalizes theorems of Halpern, Blass, and Keremedis. Secondly, we prove that the assertion that every vector space over ℤ 2 has a basis implies that every well‐ordered collection of two‐element sets has a choice function. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)