A kind of system of proof named after the mathematician David Hilbert (1862–1943). It is a precise version of the traditional notion of an axiom system. This concerns sentences of a well defined language, $ℒ$. The system comprises a decidable set of sentences of $ℒ$ called axioms, and a decidable set of rules which can be applied effectively to a finite set of sentences of $ℒ$ to deliver another. Thus, two axioms might be $p$ and $p\to q$, and a rule might be: given any sentences of the form $\phi$ and $\phi \to \psi$, infer $\psi$ (modus ponens). This rule, when applied to the two axioms, gives the sentence $q$. A proof is a series of formulae defined by recursion:

• • Any axiom is a proof.

• • Given any proof, if a rule can be applied to some of its members to deliver another sentence, the series obtained by adding this sentence to the end of the series is a proof.

A theorem is any sentence that occurs at the end of a proof. Thus, suppose that the axioms are: $p,p\to q,r,r\to s$, and the rules are modus ponens and adjunction: from formulae of the form $\phi$ and $\psi$, infer $\phi \wedge \psi$. Then the following is a proof of the theorem $q\wedge s$: