Summary: An abstract simplicial complex is a finite family of subsets of a finite set, closed under subsets. Every abstract simplicial complex \(\mathcal{C}\) naturally determines a Bratteli diagram and a stable AF-algebra \(A(\mathcal{C})\). Consider the following problem: INPUT: a pair of abstract simplicial complexes \(\mathcal{C}\) and \(\mathcal{C}'\); QUESTION: is \(A(\mathcal{C})\) isomorphic to \(A(\mathcal{C}')\)? We show that this problem is Gödel incomplete, i.e., it is recursively enumerable but not decidable. This result is in sharp contrast with the recent decidability result by \textit{O.~Bratteli, P.~E.~Jørgensen, K.~H.~Kim} and \textit{F.~Roush} [Ergodic Theory Dyn. Syst. 21, 1625-1655 (2001); corrigendum ibid. 22, 633 (2002; Zbl 1007.46046)], for the isomorphism problem of stable AF-algebras arising from the iteration of the same positive integer matrix. For the proof we use a combinatorial variant of the De~Concini-Procesi theorem for toric varieties, together with the Baker-Beynon duality theory for lattice-ordered abelian groups, Markov's undecidability result, and Elliott's classification theory for AF-algebras.