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Abstract linear dependence. (English) Zbl 0920.03014
Summary: This paper is concerned with topics in the foundations of mathematics. An important problem of an axiomatic theory is to establish the minimum number of axioms which are needed to obtain some standard properties.
In this paper we introduce and study two classes: the class of $$D$$-spaces and the class of $$DL$$-spaces. The class of $$D$$-spaces contains the class of linear vector spaces, the class of affine spaces and the class of projective spaces. The class of $$DL$$-spaces, which is contained in that of $$D$$-spaces, contains the linear vector spaces and the projective spaces. The main purpose of the paper is to prove that, starting with three axioms, in the case of the $$D$$-spaces, or four axioms, in the case of the $$DL$$-spaces, we can obtain the basic properties of linear vector spaces.
##### MSC:
 03B30 Foundations of classical theories (including reverse mathematics) 15A99 Basic linear algebra
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