Finite axiomatizability in Łukasiewicz logic

We classify every finitely axiomatizable theory in infinite-valued propositional Łukasiewicz logic by an abstract simplicial complex (V,Σ) equipped with a weight function ω:V→{1,2,…}. Using the Włodarczyk-Morelli solution of the weak Oda conjecture for toric varieties, we then construct a Turing computable one-one correspondence between (Alexander) equivalence classes of weighted abstract simplicial complexes, and equivalence classes of finitely axiomatizable theories, two theories being equivalent if their Lindenbaum algebras are isomorphic. We discuss the relationship between our classification and Markov's undecidability theorem for PL-homeomorphism of rational polyhedra.

► Finitely axiomatizable Łukasiewicz theories are classified by rational polyhedra. ► Rational polyhedra are classified by weighted abstract simplicial complexes. ► Regular triangulations of rational polyhedra correspond to nonsingular fans.