First-order definition of exactly finitely many witnesses ($\exists x_1\cdots \exists x_k[{\bigwedge }_{i<j}{x}_i\ne x_j\wedge {\bigwedge }_{i}R(x_i)\wedge \forall y(R(y)\to {\bigvee }_{i}y=x_i)]$)

For a fixed positive integer $k$, the displayed first-order logic sentence says that exactly $k$ distinct elements satisfy $R$. The pairwise inequalities and positive instances give at least $k$ witnesses; the universal exhaustiveness clause gives at most $k$. Additional parameters can remain fixed inside $R$. For $k=0$, use $\forall y\neg R(y)$ instead. Omitting distinctness or exhaustiveness changes exactly to a weaker counting condition.