= Finite axiomatizability with infinitely many scalar symbols
No finite <first-order theory>[theory] in the full language containing one <function symbol> for each <scalar multiplication>[scalar] of an infinite <field> can axiomatize exactly its nonzero <vector spaces>. It mentions only finitely many symbols; changing an unused <scalar multiplication>[scalar] function to violate preservation of zero leaves those finite sentences unchanged. Relative to the vector-space <axiom> schema, nonzeroness is instead the single <axiom> $\exists x(x\ne0)$. Absolute <finite axiomatizability>[finite axiomatizability] and finite additional axiomatizability must be distinguished.
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