# Consistency

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Non-contradiction of a theory

For other uses, see [Consistency (disambiguation)](/source/Consistency_(disambiguation)).

"Internal logic" redirects here. For the 2012 album by Grass Widow, see [Internal Logic (album)](/source/Internal_Logic_(album)).

In [deductive logic](/source/Deductive_logic), a **consistent** [theory](/source/Theory_(mathematical_logic)) is one that does not lead to a logical [contradiction](/source/Contradiction).[1] A theory T {\displaystyle T} is consistent if there is no [formula](/source/Formula_(mathematical_logic)) φ {\displaystyle \varphi } such that both φ {\displaystyle \varphi } and its negation ¬ φ {\displaystyle \lnot \varphi } are elements of the set of consequences of T {\displaystyle T} . Let A {\displaystyle A} be a set of [closed sentences](/source/Closed-form_expression) (informally "axioms") and ⟨ A ⟩ {\displaystyle \langle A\rangle } the set of closed sentences provable from A {\displaystyle A} under some (specified, possibly implicitly) formal deductive system. The set of axioms A {\displaystyle A} is **consistent** when there is no formula φ {\displaystyle \varphi } such that φ ∈ ⟨ A ⟩ {\displaystyle \varphi \in \langle A\rangle } and ¬ φ ∈ ⟨ A ⟩ {\displaystyle \lnot \varphi \in \langle A\rangle } . A *trivial* theory (i.e., one which proves every sentence in the language of the theory) is clearly inconsistent. Conversely, in an [explosive](/source/Principle_of_explosion) [formal system](/source/Formal_system) (e.g., classical or intuitionistic propositional or first-order logics) every inconsistent theory is trivial.[2]: 7 Consistency of a theory is a [syntactic](/source/Syntactic) notion, whose [semantic](/source/Semantic) counterpart is [satisfiability](/source/Satisfiable_theory). A theory is satisfiable if it has a [model](/source/Model_theory#First-order_logic), i.e., there exists an [interpretation](/source/Interpretation_(logic)) under which all [axioms](/source/Axiom) in the theory are true.[3] This is what *consistent* meant in traditional [Aristotelian logic](/source/Term_logic), although in contemporary mathematical logic the term *[satisfiable](/source/Satisfiable)* is used instead.

In a [sound formal system](/source/Soundness), every satisfiable theory is consistent, but the converse does not hold. If there exists a deductive system for which these semantic and syntactic definitions are equivalent for any theory formulated in a particular deductive [logic](/source/Mathematical_logic#Formal_logical_systems), the logic is called **[complete](/source/Completeness_(logic)#Refutation-completeness)**.[*[citation needed](https://en.wikipedia.org/wiki/Wikipedia:Citation_needed)*] The completeness of the [propositional calculus](/source/Propositional_calculus) was proved by [Paul Bernays](/source/Paul_Bernays) in 1918[*[citation needed](https://en.wikipedia.org/wiki/Wikipedia:Citation_needed)*][4] and [Emil Post](/source/Emil_Post) in 1921,[5] while the completeness of (first order) [predicate calculus](/source/Predicate_calculus) was proved by [Kurt Gödel](/source/Kurt_G%C3%B6del) in 1930,[6] and consistency proofs for arithmetics restricted with respect to the [induction axiom schema](/source/Mathematical_induction) were proved by Ackermann (1924), von Neumann (1927) and Herbrand (1931).[7] Stronger logics, such as [second-order logic](/source/Second-order_logic), are not complete.

A **consistency proof** is a [mathematical proof](/source/Mathematical_proof) that a particular theory is consistent.[8] The early development of mathematical [proof theory](/source/Proof_theory) was driven by the desire to provide finitary consistency proofs for all of mathematics as part of [Hilbert's program](/source/Hilbert's_program). Hilbert's program was strongly impacted by the [incompleteness theorems](/source/Incompleteness_theorems), which showed that sufficiently strong proof theories cannot prove their consistency (provided that they are consistent).

Although consistency can be proved using model theory, it is often done in a purely syntactical way, without any need to reference some model of the logic. The [cut-elimination](/source/Cut-elimination) (or equivalently the [normalization](/source/Normalization_property) of the [underlying calculus](/source/Curry-Howard) 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

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In theories of arithmetic, such as [Peano arithmetic](/source/Peano_arithmetic), there is an intricate relationship between the consistency of the theory and its [completeness](/source/Completeness_(logic)). 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](/source/Presburger_arithmetic) is an axiom system for the natural numbers under addition. It is both consistent and complete.

[Gödel's incompleteness theorems](/source/G%C3%B6del's_incompleteness_theorems) show that any sufficiently strong [recursively enumerable](/source/Recursively_enumerable) theory of arithmetic cannot be both complete and consistent. Gödel's theorem applies to the theories of [Peano arithmetic](/source/Peano_arithmetic) (PA) and [primitive recursive arithmetic](/source/Primitive_recursive_arithmetic) (PRA), but not to [Presburger arithmetic](/source/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](/source/Zermelo%E2%80%93Fraenkel_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 **relative consistency** is interesting in set theory (and in other sufficiently expressive axiomatic systems). If *T* is a [theory](/source/Theory_(mathematical_logic)) and *A* is an additional [axiom](/source/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](/source/Independence_(mathematical_logic)) of *T*.

## First-order logic

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### Notation

In the following context of [mathematical logic](/source/Mathematical_logic), the [turnstile symbol](/source/Turnstile_symbol) ⊢ {\displaystyle \vdash } means "provable from". That is, a ⊢ b {\displaystyle a\vdash b} reads: *b* is provable from *a* (in some specified formal system).

### Definition

- A set of [formulas](/source/Well-formed_formula) Φ {\displaystyle \Phi } in first-order logic is **consistent** (written Con ⁡ Φ {\displaystyle \operatorname {Con} \Phi } ) if there is no formula φ {\displaystyle \varphi } such that Φ ⊢ φ {\displaystyle \Phi \vdash \varphi } and Φ ⊢ ¬ φ {\displaystyle \Phi \vdash \lnot \varphi } . Otherwise Φ {\displaystyle \Phi } is **inconsistent** (written Inc ⁡ Φ {\displaystyle \operatorname {Inc} \Phi } ).

- Φ {\displaystyle \Phi } is said to be **simply consistent** if for no formula φ {\displaystyle \varphi } of Φ {\displaystyle \Phi } , both φ {\displaystyle \varphi } and the [negation](/source/Negation) of φ {\displaystyle \varphi } are theorems of Φ {\displaystyle \Phi } .[*[clarification needed](https://en.wikipedia.org/wiki/Wikipedia:Please_clarify)*]

- Φ {\displaystyle \Phi } is said to be **absolutely consistent** or **Post consistent** if at least one formula in the language of Φ {\displaystyle \Phi } is not a theorem of Φ {\displaystyle \Phi } .

- Φ {\displaystyle \Phi } is said to be **maximally consistent** if Φ {\displaystyle \Phi } is consistent and for every formula φ {\displaystyle \varphi } , Con ⁡ ( Φ ∪ { φ } ) {\displaystyle \operatorname {Con} (\Phi \cup \{\varphi \})} implies φ ∈ Φ {\displaystyle \varphi \in \Phi } .

- Φ {\displaystyle \Phi } is said to **contain witnesses** if for every formula of the form ∃ x φ {\displaystyle \exists x\,\varphi } there exists a [term](/source/Term_(logic)) t {\displaystyle t} such that ( ∃ x φ → φ t x ) ∈ Φ {\displaystyle (\exists x\,\varphi \to \varphi {t \over x})\in \Phi } , where φ t x {\displaystyle \varphi {t \over x}} denotes the [substitution](/source/Substitution_(logic)) of each x {\displaystyle x} in φ {\displaystyle \varphi } by a t {\displaystyle t} ; see also [First-order logic](/source/First-order_logic).[*[citation needed](https://en.wikipedia.org/wiki/Wikipedia:Citation_needed)*]

### Basic results

1. The following are equivalent: 1. Inc ⁡ Φ {\displaystyle \operatorname {Inc} \Phi } 1. For all φ , Φ ⊢ φ . {\displaystyle \varphi ,\;\Phi \vdash \varphi .}

1. Every satisfiable set of formulas is consistent, where a set of formulas Φ {\displaystyle \Phi } is satisfiable if and only if there exists a model I {\displaystyle {\mathfrak {I}}} such that I ⊨ Φ {\displaystyle {\mathfrak {I}}\vDash \Phi } .

1. For all Φ {\displaystyle \Phi } and φ {\displaystyle \varphi } : 1. if not Φ ⊢ φ {\displaystyle \Phi \vdash \varphi } , then Con ⁡ ( Φ ∪ { ¬ φ } ) {\displaystyle \operatorname {Con} \left(\Phi \cup \{\lnot \varphi \}\right)} ; 1. if Con ⁡ Φ {\displaystyle \operatorname {Con} \Phi } and Φ ⊢ φ {\displaystyle \Phi \vdash \varphi } , then Con ⁡ ( Φ ∪ { φ } ) {\displaystyle \operatorname {Con} \left(\Phi \cup \{\varphi \}\right)} ; 1. if Con ⁡ Φ {\displaystyle \operatorname {Con} \Phi } , then Con ⁡ ( Φ ∪ { φ } ) {\displaystyle \operatorname {Con} \left(\Phi \cup \{\varphi \}\right)} or Con ⁡ ( Φ ∪ { ¬ φ } ) {\displaystyle \operatorname {Con} \left(\Phi \cup \{\lnot \varphi \}\right)} .

1. Let Φ {\displaystyle \Phi } be a maximally consistent set of formulas and suppose it contains [witnesses](/source/Witness_(mathematics)). For all φ {\displaystyle \varphi } and ψ {\displaystyle \psi } : 1. if Φ ⊢ φ {\displaystyle \Phi \vdash \varphi } , then φ ∈ Φ {\displaystyle \varphi \in \Phi } , 1. either φ ∈ Φ {\displaystyle \varphi \in \Phi } or ¬ φ ∈ Φ {\displaystyle \lnot \varphi \in \Phi } , 1. ( φ ∨ ψ ) ∈ Φ {\displaystyle (\varphi \lor \psi )\in \Phi } if and only if φ ∈ Φ {\displaystyle \varphi \in \Phi } or ψ ∈ Φ {\displaystyle \psi \in \Phi } , 1. if ( φ → ψ ) ∈ Φ {\displaystyle (\varphi \to \psi )\in \Phi } and φ ∈ Φ {\displaystyle \varphi \in \Phi } , then ψ ∈ Φ {\displaystyle \psi \in \Phi } , 1. ∃ x φ ∈ Φ {\displaystyle \exists x\,\varphi \in \Phi } if and only if there is a term t {\displaystyle t} such that φ t x ∈ Φ {\displaystyle \varphi {t \over x}\in \Phi } .[*[citation needed](https://en.wikipedia.org/wiki/Wikipedia:Citation_needed)*]

### Henkin's theorem

Let S {\displaystyle S} be a [set of symbols](/source/Signature_(logic)). Let Φ {\displaystyle \Phi } be a maximally consistent set of S {\displaystyle S} -formulas containing [witnesses](/source/Witness_(mathematics)#Henkin_witnesses).

Define an [equivalence relation](/source/Equivalence_relation) ∼ {\displaystyle \sim } on the set of S {\displaystyle S} -terms by t 0 ∼ t 1 {\displaystyle t_{0}\sim t_{1}} if t 0 ≡ t 1 ∈ Φ {\displaystyle \;t_{0}\equiv t_{1}\in \Phi } , where ≡ {\displaystyle \equiv } denotes [equality](/source/First-order_logic#Equality_and_its_axioms). Let t ¯ {\displaystyle {\overline {t}}} denote the [equivalence class](/source/Equivalence_class) of terms containing t {\displaystyle t} ; and let T Φ := { t ¯ ∣ t ∈ T S } {\displaystyle T_{\Phi }:=\{\;{\overline {t}}\mid t\in T^{S}\}} where T S {\displaystyle T^{S}} is the set of terms based on the set of symbols S {\displaystyle S} .

Define the S {\displaystyle S} -[structure](/source/Structure_(mathematical_logic)) T Φ {\displaystyle {\mathfrak {T}}_{\Phi }} over T Φ {\displaystyle T_{\Phi }} , also called the **term-structure** corresponding to Φ {\displaystyle \Phi } , by:

1. for each n {\displaystyle n} -ary relation symbol R ∈ S {\displaystyle R\in S} , define R T Φ t 0 ¯ … t n − 1 ¯ {\displaystyle R^{{\mathfrak {T}}_{\Phi }}{\overline {t_{0}}}\ldots {\overline {t_{n-1}}}} if R t 0 … t n − 1 ∈ Φ ; {\displaystyle \;Rt_{0}\ldots t_{n-1}\in \Phi ;} [9]

1. for each n {\displaystyle n} -ary function symbol f ∈ S {\displaystyle f\in S} , define f T Φ ( t 0 ¯ … t n − 1 ¯ ) := f t 0 … t n − 1 ¯ ; {\displaystyle f^{{\mathfrak {T}}_{\Phi }}({\overline {t_{0}}}\ldots {\overline {t_{n-1}}}):={\overline {ft_{0}\ldots t_{n-1}}};}

1. for each constant symbol c ∈ S {\displaystyle c\in S} , define c T Φ := c ¯ . {\displaystyle c^{{\mathfrak {T}}_{\Phi }}:={\overline {c}}.}

Define a variable assignment β Φ {\displaystyle \beta _{\Phi }} by β Φ ( x ) := x ¯ {\displaystyle \beta _{\Phi }(x):={\bar {x}}} for each variable x {\displaystyle x} . Let I Φ := ( T Φ , β Φ ) {\displaystyle {\mathfrak {I}}_{\Phi }:=({\mathfrak {T}}_{\Phi },\beta _{\Phi })} be the **term [interpretation](/source/Interpretation_(logic)#First-order_logic)** associated with Φ {\displaystyle \Phi } .

Then for each S {\displaystyle S} -formula φ {\displaystyle \varphi } :

              I

            Φ

        ⊨
        φ

    {\displaystyle {\mathfrak {I}}_{\Phi }\vDash \varphi }

 if and only if

        φ
        ∈
        Φ
        .

    {\displaystyle \;\varphi \in \Phi .}

[*[citation needed](https://en.wikipedia.org/wiki/Wikipedia:Citation_needed)*]

### Sketch of proof

There are several things to verify. First, that ∼ {\displaystyle \sim } 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 ∼ {\displaystyle \sim } is an equivalence relation and also requires a proof that (1) and (2) are independent of the choice of t 0 , … , t n − 1 {\displaystyle t_{0},\ldots ,t_{n-1}} class representatives. Finally, I Φ ⊨ φ {\displaystyle {\mathfrak {I}}_{\Phi }\vDash \varphi } can be verified by induction on formulas.

## Model theory

In [ZFC set theory](/source/Zermelo%E2%80%93Fraenkel_set_theory) with classical [first-order logic](/source/First-order_logic),[10] an **inconsistent** theory T {\displaystyle T} is one such that there exists a closed sentence φ {\displaystyle \varphi } such that T {\displaystyle T} contains both φ {\displaystyle \varphi } and its negation φ ′ {\displaystyle \varphi '} . A **consistent** theory is one such that the following [logically equivalent](/source/Logical_equivalence) conditions hold

1. { φ , φ ′ } ⊈ T {\displaystyle \{\varphi ,\varphi '\}\not \subseteq T} [11]

1. φ ′ ∉ T ∨ φ ∉ T {\displaystyle \varphi '\not \in T\lor \varphi \not \in T}

## See also

- [Philosophy portal](https://en.wikipedia.org/wiki/Portal:Philosophy)

Wikiquote has quotations related to ***[Consistency](https://en.wikiquote.org/wiki/Special:Search/Consistency)***.

- [Cognitive dissonance](/source/Cognitive_dissonance) – Mental phenomenon of holding contradictory beliefs

- [Equiconsistency](/source/Equiconsistency) – Being equally consistent

- [Hilbert's problems](/source/Hilbert's_problems) – 23 mathematical problems stated in 1900

- [Hilbert's second problem](/source/Hilbert's_second_problem) – Consistency of the axioms of arithmetic

- [Jan Łukasiewicz](/source/Jan_%C5%81ukasiewicz) – Polish logician and philosopher (1878–1956)

- [Paraconsistent logic](/source/Paraconsistent_logic) – Type of formal logic

- [ω-consistency](/source/%CE%A9-consistency) – Mathematical theoryPages displaying short descriptions of redirect targets

- [Gentzen's consistency proof](/source/Gentzen's_consistency_proof) – Mathematical logic concept

- [Proof by contradiction](/source/Proof_by_contradiction) – Form of proof

- [Self-refuting idea](/source/Self-refuting_idea) – Idea that refutes itself

## Notes

1. **[^](#cite_ref-1)** [Tarski 1946](#CITEREFTarski1946) states it this way: "A deductive theory is called *consistent* or *non-contradictory* if no two asserted statements of this theory contradict each other, or in other words, if of any two contradictory sentences … at least one cannot be proved," (p. 135) where Tarski defines *contradictory* as follows: "With the help of the word *not* one forms the *negation* of any sentence; two sentences, of which the first is a negation of the second, are called *contradictory sentences*" (p. 20). This definition requires a notion of "proof". [Gödel 1931](#CITEREFGödel1931) defines the notion this way: "The class of *provable formulas* is defined to be the smallest class of formulas that contains the axioms and is closed under the relation "immediate consequence", i.e., formula *c* of *a* and *b* is defined as an *immediate consequence* in terms of *modus ponens* or substitution; cf [Gödel 1931](#CITEREFGödel1931), [van Heijenoort 1967](#CITEREFvan_Heijenoort1967), p. 601. Tarski defines "proof" informally as "statements follow one another in a definite order according to certain principles … and accompanied by considerations intended to establish their validity [true conclusion] for all true premises – [Reichenbach 1947](#CITEREFReichenbach1947), p. 68]" cf [Tarski 1946](#CITEREFTarski1946), p. 3. [Kleene 1952](#CITEREFKleene1952) defines the notion with respect to either an induction or as to paraphrase) a finite sequence of formulas such that each formula in the sequence is either an axiom or an "immediate consequence" of the preceding formulas; "A *proof is said to be a proof*of*its last formula, and this formula is said to be*(formally) provable*or be a*(formal) theorem" cf [Kleene 1952](#CITEREFKleene1952), p. 83.

1. **[^](#cite_ref-Carnielli_2-0)** Carnielli, Walter; Coniglio, Marcelo Esteban (2016). *Paraconsistent logic: consistency, contradiction and negation*. Logic, Epistemology, and the Unity of Science. Vol. 40. Cham: Springer. [doi](/source/Doi_(identifier)):[10.1007/978-3-319-33205-5](https://doi.org/10.1007%2F978-3-319-33205-5). [ISBN](/source/ISBN_(identifier)) [978-3-319-33203-1](https://en.wikipedia.org/wiki/Special:BookSources/978-3-319-33203-1). [MR](/source/MR_(identifier)) [3822731](https://mathscinet.ams.org/mathscinet-getitem?mr=3822731). [Zbl](/source/Zbl_(identifier)) [1355.03001](https://zbmath.org/?format=complete&q=an:1355.03001).

1. **[^](#cite_ref-3)** Hodges, Wilfrid (1997). *A Shorter Model Theory*. New York: Cambridge University Press. p. 37. Let L {\displaystyle L} be a signature, T {\displaystyle T} a theory in L ∞ ω {\displaystyle L_{\infty \omega }} and φ {\displaystyle \varphi } a sentence in L ∞ ω {\displaystyle L_{\infty \omega }} . We say that φ {\displaystyle \varphi } is a *consequence* of T {\displaystyle T} , or that T {\displaystyle T} *entails* φ {\displaystyle \varphi } , in symbols T ⊢ φ {\displaystyle T\vdash \varphi } , if every model of T {\displaystyle T} is a model of φ {\displaystyle \varphi } . (In particular if T {\displaystyle T} has no models then T {\displaystyle T} entails φ {\displaystyle \varphi } .) *Warning*: we don't require that if T ⊢ φ {\displaystyle T\vdash \varphi } then there is a proof of φ {\displaystyle \varphi } from T {\displaystyle T} . In any case, with infinitary languages, it's not always clear what would constitute proof. Some writers use T ⊢ φ {\displaystyle T\vdash \varphi } to mean that φ {\displaystyle \varphi } is deducible from T {\displaystyle T} in some particular formal proof calculus, and they write T ⊨ φ {\displaystyle T\models \varphi } for our notion of entailment (a notation which clashes with our A ⊨ φ {\displaystyle A\models \varphi } ). For first-order logic, the two kinds of entailment coincide by the completeness theorem for the proof calculus in question. We say that φ {\displaystyle \varphi } is *valid*, or is a *logical theorem*, in symbols ⊢ φ {\displaystyle \vdash \varphi } , if φ {\displaystyle \varphi } is true in every L {\displaystyle L} -structure. We say that φ {\displaystyle \varphi } is *consistent* if φ {\displaystyle \varphi } is true in some L {\displaystyle L} -structure. Likewise, we say that a theory T {\displaystyle T} is *consistent* if it has a model. We say that two theories S and T in L infinity omega are equivalent if they have the same models, i.e. if Mod(S) = Mod(T). (Please note the definition of Mod(T) on p. 30 ...)

1. **[^](#cite_ref-4)** [van Heijenoort 1967](#CITEREFvan_Heijenoort1967), p. 265 states that Bernays determined the *independence* of the axioms of *Principia Mathematica*, a result not published until 1926, but he says nothing about Bernays proving their *consistency*.

1. **[^](#cite_ref-5)** Post proves both consistency and completeness of the propositional calculus of PM, cf van Heijenoort's commentary and Post's 1931 *Introduction to a general theory of elementary propositions* in [van Heijenoort 1967](#CITEREFvan_Heijenoort1967), pp. 264ff. Also [Tarski 1946](#CITEREFTarski1946), pp. 134ff.

1. **[^](#cite_ref-6)** cf van Heijenoort's commentary and Gödel's 1930 *The completeness of the axioms of the functional calculus of logic* in [van Heijenoort 1967](#CITEREFvan_Heijenoort1967), pp. 582ff.

1. **[^](#cite_ref-7)** cf van Heijenoort's commentary and Herbrand's 1930 *On the consistency of arithmetic* in [van Heijenoort 1967](#CITEREFvan_Heijenoort1967), pp. 618ff.

1. **[^](#cite_ref-8)** A consistency proof often assumes the consistency of another theory. In most cases, this other theory is [Zermelo–Fraenkel set theory](/source/Zermelo%E2%80%93Fraenkel_set_theory) with or without the [axiom of choice](/source/Axiom_of_choice) (this is equivalent since these two theories have been proved equiconsistent; that is, if one is consistent, the same is true for the other).

1. **[^](#cite_ref-9)** This definition is independent of the choice of t i {\displaystyle t_{i}} due to the substitutivity properties of ≡ {\displaystyle \equiv } and the maximal consistency of Φ {\displaystyle \Phi } .

1. **[^](#cite_ref-10)** the common case in many applications to other areas of mathematics as well as the ordinary mode of reasoning of [informal mathematics](/source/Informal_mathematics) in calculus and applications to physics, chemistry, engineering

1. **[^](#cite_ref-11)** according to [De Morgan's laws](/source/De_Morgan's_laws)

## References

- Gödel, Kurt (1 December 1931). "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I". *Monatshefte für Mathematik und Physik*. **38** (1): 173–198. [doi](/source/Doi_(identifier)):[10.1007/BF01700692](https://doi.org/10.1007%2FBF01700692).

- [Kleene, Stephen](/source/Stephen_Kleene) (1952). *Introduction to Metamathematics*. New York: North-Holland. [ISBN](/source/ISBN_(identifier)) [0-7204-2103-9](https://en.wikipedia.org/wiki/Special:BookSources/0-7204-2103-9). {{[cite book](https://en.wikipedia.org/wiki/Template:Cite_book)}}: ISBN / Date incompatibility ([help](https://en.wikipedia.org/wiki/Help:CS1_errors#invalid_isbn_date)) 10th impression 1991.

- [Reichenbach, Hans](/source/Hans_Reichenbach) (1947). *Elements of Symbolic Logic*. New York: Dover. [ISBN](/source/ISBN_(identifier)) [0-486-24004-5](https://en.wikipedia.org/wiki/Special:BookSources/0-486-24004-5). {{[cite book](https://en.wikipedia.org/wiki/Template:Cite_book)}}: ISBN / Date incompatibility ([help](https://en.wikipedia.org/wiki/Help:CS1_errors#invalid_isbn_date))

- [Tarski, Alfred](/source/Alfred_Tarski) (1946). *Introduction to Logic and to the Methodology of Deductive Sciences* (Second ed.). New York: Dover. [ISBN](/source/ISBN_(identifier)) [0-486-28462-X](https://en.wikipedia.org/wiki/Special:BookSources/0-486-28462-X). {{[cite book](https://en.wikipedia.org/wiki/Template:Cite_book)}}: ISBN / Date incompatibility ([help](https://en.wikipedia.org/wiki/Help:CS1_errors#invalid_isbn_date))

- [van Heijenoort, Jean](/source/Jean_van_Heijenoort) (1967). *From Frege to Gödel: A Source Book in Mathematical Logic*. Cambridge, MA: Harvard University Press. [ISBN](/source/ISBN_(identifier)) [0-674-32449-8](https://en.wikipedia.org/wiki/Special:BookSources/0-674-32449-8). (pbk.)

- "Consistency". *[The Cambridge Dictionary of Philosophy](/source/The_Cambridge_Dictionary_of_Philosophy)*.

- Ebbinghaus, H. D.; Flum, J.; Thomas, W. *Mathematical Logic*.

- Jevons, W. S. (1870). *Elementary Lessons in Logic*.

## External links

Look up ***[consistency](https://en.wiktionary.org/wiki/Special:Search/consistency)*** in Wiktionary, the free dictionary.

- Mortensen, Chris (2017). ["Inconsistent Mathematics"](http://plato.stanford.edu/entries/mathematics-inconsistent/). *[Stanford Encyclopedia of Philosophy](/source/Stanford_Encyclopedia_of_Philosophy)*.

v t e ‌Logical truth ⊤ Functional: Truth value Truth function ⊨ Tautology Formal: Theory Formal proof Theorem Negation ⊥ False Contradiction Inconsistency

v t e Mathematical logic General Axiom list Cardinality First-order logic Formal proof Formal semantics Foundations of mathematics Information theory Lemma Logical consequence Model Theorem Theory Type theory Theorems (list), paradoxes Gödel's completeness – incompleteness theorems Tarski's undefinability Banach–Tarski paradox Cantor's theorem – paradox – diagonal argument Compactness Halting problem Lindström's Löwenheim–Skolem Russell's paradox Logics Traditional Classical logic Logical truth Tautology Proposition Inference Logical equivalence Consistency Equiconsistency Argument Soundness Validity Syllogism Square of opposition Venn diagram Propositional Boolean algebra Boolean functions Logical connectives Propositional calculus Propositional formula Truth tables Many-valued logic 3 finite ∞ Predicate First-order list Second-order Monadic Higher-order Fixed-point Free Quantifiers Predicate Monadic predicate calculus Set theory Set hereditary Class (Ur-)Element Ordinal number Extensionality Forcing Relation equivalence partition Set operations: intersection union complement Cartesian product power set identities Types of sets Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter Recursive Fuzzy Universal Universe constructible Grothendieck Von Neumann Maps, cardinality Function/Map domain codomain image In/Sur/Bi-jection Schröder–Bernstein theorem Isomorphism Gödel numbering Enumeration Large cardinal inaccessible Aleph number Operation binary Theories Zermelo–Fraenkel axiom of choice continuum hypothesis General Kripke–Platek Morse–Kelley Naive New Foundations Tarski–Grothendieck Von Neumann–Bernays–Gödel Ackermann Constructive Formal systems (list), language, syntax Alphabet Arity Automata Axiom schema Expression ground Extension by definition conservative Relation Formation rule Grammar Formula atomic closed ground open Free/bound variable Language Metalanguage Logical connective ¬ ∨ ∧ → ↔ = Predicate functional variable propositional variable Proof Quantifier ∃ ! ∀ rank Sentence atomic spectrum Signature String Substitution Symbol function logical/constant non-logical variable Term Theory list Example axiomatic systems (list) of true arithmetic Peano second-order elementary function primitive recursive Robinson Skolem of the real numbers Tarski's axiomatization of Boolean algebras canonical minimal axioms of geometry Euclidean Elements Hilbert's Tarski's non-Euclidean Principia Mathematica Proof theory Formal proof Natural deduction Logical consequence Rule of inference Sequent calculus Theorem Systems axiomatic deductive Hilbert list Complete theory Independence (from ZFC) Proof of impossibility Ordinal analysis Reverse mathematics Self-verifying theories Model theory Interpretation function of models Model atomic equivalence finite prime saturated spectrum submodel Non-standard model of non-standard arithmetic Diagram elementary Categorical theory Model complete theory Satisfiability Semantics of logic Strength Theories of truth semantic Tarski's Kripke's T-schema Transfer principle Truth predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable set Decision problem decidable undecidable P NP P versus NP problem Kolmogorov complexity Lambda calculus Primitive recursive function Recursion Recursive set Turing machine Type theory Related Abstract logic Algebraic logic Automated theorem proving Category theory Concrete/Abstract category Category of sets History of logic History of mathematical logic timeline Logicism Mathematical object Philosophy of mathematics Supertask Mathematics portal

v t e Metalogic and metamathematics Cantor's theorem Entscheidungsproblem Church–Turing thesis Consistency Effective method Foundations of mathematics of geometry Gödel's completeness theorem Gödel's incompleteness theorems Soundness Completeness Decidability Interpretation Löwenheim–Skolem theorem Metatheorem Satisfiability Independence Type–token distinction Use–mention distinction

Authority control databases GND

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Adapted from the Wikipedia article [Consistency](https://en.wikipedia.org/wiki/Consistency) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Consistency?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.