The class of all Artinian local rings of length at most l is ∀2-elementary, axiomatised by a finite set of axioms Artl. We show that its existentially closed models are Gorenstein, of length exactly l and their residue fields are algebraically closed, and, conversely, every existentially closed model is of this form. The theory Gorl of all Artinian local Gorenstein rings of length l with algebr...