Bases, spanning sets, and the axiom of choice | Zendy

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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)

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