= Independence of a pair is not definable over an infinite field
In the one-sorted vector-space language naming each <scalar multiplication>, a <first-order formula> reduces by <quantifier elimination> to finitely many <linear equations>. An independent pair satisfies only the identically zero equations among them. A dependent pair $(x,ax)$ with nonzero $x$ satisfies the same listed equations if the <scalar multiplication>[scalar] $a$ avoids their finitely many forbidden ratios, which is possible over an infinite <field>. Thus no formula defines pair independence. The complete independent-pair type is still <consistent first-order theory>[consistent] in an infinite-dimensional <first-order model>[model]; it is nonisolated.
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