Gödel’s theorem applies to the theories of Peano arithmetic (PA) and primitive recursive arithmetic (PRA), but not to Presburger arithmetic. A consistency proof is a mathematical proof that a particular theory is consistent.[7] 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 consistency (provided that they are consistent). 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. 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).

These examples are programmatically compiled from various online sources to illustrate current usage of the word ‘consistency.’ Any opinions expressed in the examples do not represent those of Merriam-Webster or its editors.

## consistency American Dictionary

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. These examples are programmatically compiled from various online sources to illustrate current usage of the word ‘consistent.’ Any opinions expressed in the examples do not represent those of Merriam-Webster or its editors.

- These examples are programmatically compiled from various online sources to illustrate current usage of the word ‘consistency.’ Any opinions expressed in the examples do not represent those of Merriam-Webster or its editors.
- 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.
- Hilbert’s program was strongly impacted by the incompleteness theorems, which showed that sufficiently strong proof theories cannot prove their consistency (provided that they are consistent).
- These examples are programmatically compiled from various online sources to illustrate current usage of the word ‘consistent.’ Any opinions expressed in the examples do not represent those of Merriam-Webster or its editors.

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. Gödel’s incompleteness theorems show that any sufficiently strong recursively enumerable theory of arithmetic cannot be both complete and consistent.