# Consistency

### From Logic

In logic, a **consistent** 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 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.** 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. 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 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 obviously no cut-free proof of falsity, there is no contradiction in general.

## Contents |

## Consistency and completeness in arithmetic

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.

## Formulas

A set of formulas **Failed to parse (Can't write to or create math temp directory): \\Phi**

in first-order logic isconsistent(written ConFailed to parse (Can't write to or create math temp directory): \\Phi

) if and only if there is no formula **Failed to parse (Can't write to or create math temp directory): \\phi**

such thatFailed to parse (Can't write to or create math temp directory): \\Phi \\vdash \\phiandFailed to parse (Can't write to or create math temp directory): \\Phi \\vdash \\lnot\\phi

. Otherwise **Failed to parse (Can't write to or create math temp directory): \\Phi**

isinconsistentand is written IncFailed to parse (Can't write to or create math temp directory): \\Phi

.

**Failed to parse (Can't write to or create math temp directory): \\Phi**

is said to besimply consistentif and only if for no formulaFailed to parse (Can't write to or create math temp directory): \\phiofFailed to parse (Can't write to or create math temp directory): \\Phiare bothFailed to parse (Can't write to or create math temp directory): \\phiand the negation ofFailed to parse (Can't write to or create math temp directory): \\phitheorems ofFailed to parse (Can't write to or create math temp directory): \\Phi

.

**Failed to parse (Can't write to or create math temp directory): \\Phi**

is said to beabsolutely consistentorPost consistentif and only if at least one formula ofFailed to parse (Can't write to or create math temp directory): \\Phiis not a theorem ofFailed to parse (Can't write to or create math temp directory): \\Phi

.

**Failed to parse (Can't write to or create math temp directory): \\Phi**

is said to bemaximally consistentif and only if for every formulaFailed to parse (Can't write to or create math temp directory): \\phi

, if Con **Failed to parse (Can't write to or create math temp directory): \\Phi \\cup \\phi**

thenFailed to parse (Can't write to or create math temp directory): \\phi \\in \\Phi

.

**Failed to parse (Can't write to or create math temp directory): \\Phi**

is said tocontain witnessesif and only if for every formula of the formFailed to parse (Can't write to or create math temp directory): \\exists x \\phithere exists a termFailed to parse (Can't write to or create math temp directory): tsuch thatFailed to parse (Can't write to or create math temp directory): (\\exists x \\phi \\to \\phi {t \\over x}) \\in \\Phi

. See First-order logic.

### Basic results

**1.** The following are equivalent:

(a) Inc**Failed to parse (Can't write to or create math temp directory): \\Phi**

(b) For all **Failed to parse (Can't write to or create math temp directory): \\phi,\\; \\Phi \\vdash \\phi.**

**2.** Every satisfiable set of formulas is consistent, where a set of formulas **Failed to parse (Can't write to or create math temp directory): \\Phi**

is satisfiable if and only if there exists a modelFailed to parse (Can't write to or create math temp directory): \\mathfrak{I}such thatFailed to parse (Can't write to or create math temp directory): \\mathfrak{I} \\vDash \\Phi

.

**3.** For all **Failed to parse (Can't write to or create math temp directory): \\Phi**

andFailed to parse (Can't write to or create math temp directory): \\phi

(a) if not **Failed to parse (Can't write to or create math temp directory): \\Phi \\vdash \\phi**
, then Con**Failed to parse (Can't write to or create math temp directory): \\Phi \\cup \\{\\lnot\\phi\\}**

(b) if Con **Failed to parse (Can't write to or create math temp directory): \\Phi**

andFailed to parse (Can't write to or create math temp directory): \\Phi \\vdash \\phi

, then Con**Failed to parse (Can't write to or create math temp directory): \\Phi \\cup \\{\\phi\\}**

(c) if Con **Failed to parse (Can't write to or create math temp directory): \\Phi**
, then Con**Failed to parse (Can't write to or create math temp directory): \\Phi \\cup \\{\\phi\\}**

or ConFailed to parse (Can't write to or create math temp directory): \\Phi \\cup \\{\\lnot \\phi\\}

.

**4.** Let **Failed to parse (Can't write to or create math temp directory): \\Phi**

be a maximally consistent set of formulas and contain witnesses. For allFailed to parse (Can't write to or create math temp directory): \\phiandFailed to parse (Can't write to or create math temp directory): \\psi

(a) if **Failed to parse (Can't write to or create math temp directory): \\Phi \\vdash \\phi**
, then **Failed to parse (Can't write to or create math temp directory): \\phi \\in \\Phi**
,

(b) either **Failed to parse (Can't write to or create math temp directory): \\phi \\in \\Phi**

orFailed to parse (Can't write to or create math temp directory): \\lnot \\phi \\in \\Phi

,

(c) **Failed to parse (Can't write to or create math temp directory): (\\phi \\or \\psi) \\in \\Phi**

if and only ifFailed to parse (Can't write to or create math temp directory): \\phi \\in \\PhiorFailed to parse (Can't write to or create math temp directory): \\psi \\in \\Phi

,

(d) if **Failed to parse (Can't write to or create math temp directory): (\\phi\\to\\psi) \\in \\Phi**

andFailed to parse (Can't write to or create math temp directory): \\phi \\in \\Phi

, then **Failed to parse (Can't write to or create math temp directory): \\psi \\in \\Phi**
,

(e) **Failed to parse (Can't write to or create math temp directory): \\exists x \\phi \\in \\Phi**

if and only if there is a termFailed to parse (Can't write to or create math temp directory): tsuch thatFailed to parse (Can't write to or create math temp directory): \\phi{t \\over x}\\in\\Phi

.

## References

- The Cambridge Dictionary of Philosophy,
*consistency* - H.D. Ebbinghaus, J. Flum, W. Thomas,
**Mathematical Logic** - Jevons, W.S.,
*Elementary Lessons in Logic, 1870*