## Consistency |

In **consistent** ^{[1]}^{[2]} 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 **satisfiable** is used instead. The syntactic definition states a theory is consistent if there is no **consistent** when for no formula .^{[3]}

If there exists a deductive system for which these semantic and syntactic definitions are equivalent for any theory formulated in a particular deductive ** complete**.

A **consistency proof** is a ^{[8]} The early development of mathematical

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

- consistency and completeness in arithmetic and set theory
- first-order logic
- model theory
- see also
- footnotes
- references
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In theories of arithmetic, such as

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

Because consistency of ZF is not provable in ZF, the weaker notion **relative consistency** is interesting in set theory (and in other sufficiently expressive axiomatic systems). If *T* is a *A* is an additional *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 *T*.