Consistency
In classical deductive logic, a consistent theory is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent if it has a model, i.e., there exists an interpretation under which all formulas in the theory are true. This is the sense used in traditional Aristotelian logic, although in contemporary mathematical logic the term "satisfiable" is used instead. The syntactic definition states a theory formula_1 is consistent if there is no formula formula_2 such that both formula_2 and its  negation formula_4 are elements of the set of consequences of formula_1. Let formula_6 be a set of closed sentences (informally "axioms") and formula_7 the set of closed sentences provable from formula_6 under some (specified, possibly implicitly) formal deductive system. The set of axioms formula_6 is consistent when formula_10 for no formula formula_2. If there exists a deductive system for which these semantic and syntactic definitions are equivalent for any theory formulated in a particular deductive logic, the logic is called complete. The completeness of the 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 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 the 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 of the underlying calculus if there is one) implies the consistency of the calculus: since there is no cut-free proof of falsity, there is no contradiction in general. Consistency and completeness in arithmetic and set theory. In theories of arithmetic, such as Peano arithmetic, there is an  intricate relationship between the consistency of the theory and its completeness. A theory is complete if, for every formula φ in its language, at least one of φ or ¬φ is a logical consequence of the theory. Presburger arithmetic is an axiom system for the natural numbers under addition. It is both consistent and complete. Gödel's incompleteness theorems show that any sufficiently strong recursively enumerable theory of arithmetic cannot be both complete and consistent. Gödel's theorem applies to the theories of Peano arithmetic (PA) and primitive recursive arithmetic (PRA), but not to Presburger arithmetic. Moreover, Gödel's  second incompleteness theorem shows that the consistency of sufficiently strong recursively enumerable theories of arithmetic can be tested in a particular way. Such a theory is consistent if and only if it does "not" prove a particular sentence, called the Gödel sentence of the theory, which is a formalized statement of the claim that the theory is indeed consistent. Thus the consistency of a sufficiently strong, recursively enumerable, consistent theory of arithmetic can never be proven in that system itself. The same result is true for recursively enumerable theories that can describe a strong enough fragment of arithmetic—including set  theories such as Zermelo–Fraenkel set theory (ZF). These set theories cannot prove their own Gödel sentence—provided that they are consistent, which is generally believed. Because consistency of ZF is not provable in ZF, the weaker notion is interesting in set theory (and in other sufficiently expressive axiomatic systems). If "T" is a theory and "A" is an additional axiom, "T" + "A" is said to be consistent relative to "T" (or simply that "A" is consistent with "T") if it can be proved that if "T" is consistent then "T" + "A" is consistent. If both "A  and ¬A" are consistent with "T", then "A" is said to be independent of "T". First-order logic. Notation. formula_12 (Turnstile symbol) in the following context of mathematical logic, means "provable from". That is, formula_13 reads: "b" is provable from "a" (in some specified formal system). See List of logic symbols. In other cases, the turnstile symbol may mean implies; permits the derivation of. See: List of mathematical symbols. Henkin's theorem. Let formula_74 be a set of symbols. Let formula_14 be a maximally consistent set of formula_74-formulas containing witnesses. Define an equivalence relation formula_77 on the set of  formula_74-terms by formula_79 if formula_80, where formula_81 denotes equality. Let formula_82 denote the equivalence class of terms containing formula_83; and let formula_84 where formula_85 is the set of terms based on the set of symbols formula_74. Define the formula_74-structure formula_88 over formula_89, also called the term-structure corresponding to formula_14, by: Define a variable assignment formula_100 by formula_101 for each variable formula_40. Let formula_103 be the term interpretation associated with formula_14. Then for each formula_74-formula formula_2: formula_107 if and only if formula_108 Sketch of proof. There are several things to verify. First, that formula_77 is in fact  an equivalence relation. Then, it needs to be verified that (1), (2), and (3) are well defined. This falls out of the fact that formula_77 is an equivalence relation and also requires a proof that (1) and (2) are independent of the choice of formula_111 class representatives. Finally, formula_112 can be verified by induction on formulas. Model theory. In ZFC set theory with classical first-order logic, an inconsistent theory formula_1 is one such that there exists a closed sentence formula_2 such that formula_1 contains both formula_2 and its negation formula_117. A consistent theory is one such that the following  logically equivalent conditions hold