stylistic:style_transfer
Generalizations

Using the deduction theorem, the Entscheidungsproblem encompasses the more general problem of deciding whether a given first-order sentence is entailed by a given finite set of sentences, but validity in first-order theories with infinitely many axioms cannot be directly reduced to the Entscheidungsproblem. Such more general decision problems are of practical interest. Some first-order theories are algorithmically decidable; examples of this include Presburger arithmetic, real closed fields, and static type systems of many programming languages. On the other hand, the first-order theory of the natural numbers with addition and multiplication expressed by Peano's axioms cannot be decided with an algorithm.