As I understand it, Hilbert's original solution to Gordan's Problem was nonconstructive, proving that every algebraic variety over a field had a finite generating set. (His result is now generally cited as "Hilbert's Basis Theorem", that polynomial rings over Noetherian rings are Noetherian.)
In modern day algebraic geometry, Hilbert's nonconstructive argument is replaced by a very constructive process in which one generates a Groebner Basis for the algebraic set.