On the decision problem for theories of finite models

An infinite extension of the elementary theory of Abelian groups is constructed, which is proved to be decidable, while the elementary theory of its finite models is shown to be undecidable. Tarski’s proof of undecidability for the elementary theory of Abelian cancellation semigroups is presented in detail. Szmielew’s proof of the decidability of the elementary theory of Abelian groups is used to prove the decidability of the elementary theory of finite Abelian groups, and an axiom system for this theory is exhibited. It follows that the elementary theory of Abelian cancellation semigroups, while undecidable, has a decidable theory of finite models.