In logic, a consistent theory is one that does not contain a contradiction^{[1]}. 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; this is the sense used in traditional Aristotelian logic, although in contemporary mathematical logic the term satisfiable is used instead. The syntactic definition states that a theory is consistent if there is no 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 complete.^{[2]} The completeness of sentential calculus was proved by Paul Bernays in 1918^{[3]} and Emil Post in 1921^{[4]}, while the completeness of predicate calculus was proved by Kurt Gödel in 1930^{[5]}, and consistency proofs for arithmetics restricted with respect to the induction axiom schema were proved by Ackermann (1924), von Neumann (1927) and Herbrand (1931)^{[6]}. Stronger logics, such as secondorder 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 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 cutelimination (or equivalently the normalization of the underlying calculus if there is one) implies the consistency of the calculus: since there is obviously no cutfree proof of falsity, there is no contradiction in general.
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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 effective 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 effective 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, effective, consistent theory of arithmetic can never be proven in that system itself. The same result is true for effective theories that can describe a strong enough fragment of arithmetic – including set theories such as Zermelo–Frankel set theory. These set theories cannot prove their own Gödel sentences  provided that they are consistent, which is generally believed.
A set of formulas Φ in firstorder logic is consistent (written ConΦ) if and only if there is no formula φ such that and . Otherwise Φ is inconsistent and is written IncΦ.
Φ is said to be simply consistent if and only if for no formula φ of Φ are both φ and the negation of φ theorems of Φ.
Φ is said to be absolutely consistent or Post consistent if and only if at least one formula of Φ is not a theorem of Φ.
Φ is said to be maximally consistent if and only if for every formula φ, if Con then .
Φ is said to contain witnesses if and only if for every formula of the form there exists a term t such that . See Firstorder logic.
1. The following are equivalent:
(a) IncΦ
(b) For all
2. Every satisfiable set of formulas is consistent, where a set of formulas Φ is satisfiable if and only if there exists a model such that .
3. For all Φ and φ:
(a) if not , then Con;
(b) if Con Φ and , then Con;
(c) if Con Φ, then Con or Con.
4. Let Φ be a maximally consistent set of formulas and contain witnesses. For all φ and ψ:
(a) if , then ,
(b) either or ,
(c) if and only if or ,
(d) if and , then ,
(e) if and only if there is a term t such that .
Let Φ be a maximally consistent set of formulas containing witnesses.
Define a binary relation on the set of Sterms if and only if ; and let denote the equivalence class of terms containing ; and let where is the set of terms based on the symbol set .
Define the Sstructure over the termstructure corresponding to Φ by:
(1) For nary , if and only if ,
(2) For nary , ,
(3) For , .
Let be the term interpretation associated with Φ, where .
There are several things to verify. First, that ˜ is an equivalence relation. Then, it needs to be verified that (1), (2), and (3) are well defined. This falls out of the fact that ˜ is an equivalence relation and also requires a proof that (1) and (2) are independent of the choice of class representatives. Finally, can be verified by induction on formulas.
