  We consider when the symmetric algebra of an infinite-dimensional Lie
algebra, equipped with the natural Poisson bracket, satisfies the ascending
chain condition (ACC) on Poisson ideals. We define a combinatorial condition on
a graded Lie algebra which we call Dicksonian because it is related to
Dickson's lemma on finite subsets of $\mathbb N^k$. Our main result is:
  Theorem. If $\mathfrak g$ is a Dicksonian graded Lie algebra over a field of
characteristic zero, then the symmetric algebra $S(\mathfrak g)$ satisfies the
ACC on radical Poisson ideals.
  As an application, we establish this ACC for the symmetric algebra of any
graded simple Lie algebra of polynomial growth over an algebraically closed
field of characteristic zero, and for the symmetric algebra of the Virasoro
algebra. We also derive some consequences connected to the Poisson primitive
spectrum of finitely Poisson-generated algebras.
