By googling one sees that each of the following statements has a significant number of believers:
(1) the vector space {0} has no basis,
(2) the empty set is a basis of {0} by convention,
(3) the statements "{0} has no basis" and "the empty set is a basis of {0}" are equivalent,
(4) the statements "{0} has no basis" and "the empty set is a basis of {0}" are NOT equivalent,
(5) the statement "the empty set is a basis of {0}" is an immediate consequence of the definitions of the terms involved.
I think that we'll all agree that the 5 beliefs are not ALL true. My personal religion is to believe in (4) and (5). I don't think I'll ever understand the arguments in favor of (1), (2) or (3).
