= Solution
\b[No.] If a finite <conjunction> $\phi$ axiomatized the infinite spaces, it would follow from the vector-space <axioms> together with all the $D_n$. Compactness would make it follow from finitely many of those <axioms> and <dimension of a vector space>[dimension] requirements. Let $N$ be at least every <dimension of a vector space>[dimension] mentioned there. The finite-dimensional space $K^N$ satisfies all vector-space <axioms> and those finitely many $D_n$, hence would satisfy $\phi$. But it is a <finite set> and so cannot be a <first-order model>[model] of the purported axiomatization. This <contradiction> proves failure of <finite axiomatizability>[finite axiomatizability], including <finite axiomatizability>[finite axiomatizability] relative to <vector spaces>.
Back to article page
