In logic, a consistent theory (mathematical logic)|theory is one that does not contain a contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent if and only if it has a Model theory#First-order logic|model, i.e. there exists an interpretation (logic)|interpretation under which all Well-formed formula|formulas in the theory are true. This is the sense used in traditional Term logic|Aristotelian logic, although in contemporary mathematical logic the term satisfiable is used instead. The syntactic definition states that a theory is consistent if and only if there is no Formula (mathematical logic)|formula P such that both P and its negation are provable from the axioms of the theory under its associated deductive system.
If these semantic and syntactic definitions are equivalent for a particular logic, the logic is Completeness#Logical completeness|complete. The completeness of sentential calculus was proved by Paul Bernays in 1918 and Emil Post in 1921, while the completeness of predicate calculus was proved by Kurt Gödel in 1930, and consistency proofs for arithmetics restricted with respect to the Mathematical induction|induction axiom schema were proved by Ackermann (1924), von Neumann (1927) and Herbrand (1931). Stronger logics, such as second-order logic, are not complete.
A consistency proof is a mathematical proof that a particular theory is consistent. The early development of mathematical proof theory was driven by the desire to provide finitary consistency proofs for all of mathematics as part of Hilbert's program. Hilbert's program was strongly impacted by Gödel's incompleteness theorems|incompleteness theorems, which showed that sufficiently strong proof theories cannot prove their own consistency (provided that they are in fact consistent).
Although consistency can be proved by means of model theory, it is often done in a purely syntactical way, without any need to reference some model of the logic. The cut-elimination (or equivalently the Normalization property|normalization of the Curry-Howard|underlying calculus if there is one) implies the consistency of the calculus: since there is obviously no cut-free proof of falsity, there is no contradiction in general.

In logic, a consistent theory (mathematical logic)|theory is one that does not contain a contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent if and only if it has a Model theory#First-order logic|model, i.e. there exists an interpretation (logic)|interpretation under which all Well-formed formula|formulas in the theory are true. This is the sense used in traditional Term logic|Aristotelian logic, although in contemporary mathematical logic the term satisfiable is used instead. The syntactic definition states that a theory is consistent if and only if there is no Formula (mathematical logic)|formula P such that both P and its negation are provable from the axioms of the theory under its associated deductive system.
If these semantic and syntactic definitions are equivalent for a particular logic, the logic is Completeness#Logical completeness|complete. The completeness of sentential...