Elementary properties of free lattices III: Undecidability of the full theory

In [6] we proved that the universal theory of infinite free lattices is (algorithmically) decidable, leaving open the problem of decidability of the full theory of an (infinite) free lattice. We solve this problem by proving that, for every cardinal $\kappa \ge 3$, the first-order theory of the free lattice ${\mathbf{F}}_{\kappa }$ is undecidable.