View source for Consistency

{{Short description|Non-contradiction of a theory}}
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In [[deductive logic]], a '''consistent''' [[theory (mathematical logic)|theory]] is one that does not lead to a logical [[contradiction]].<ref>{{Harvnb|Tarski|1946}} 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". {{Harvnb|Gödel|1931}} 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 {{Harvnb|Gödel|1931}}, {{Harvnb|van Heijenoort|1967|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 – {{Harvnb|Reichenbach|1947|p=68}}]" cf {{Harvnb|Tarski|1946|p=3}}. {{Harvnb|Kleene|1952}} 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 {{harvnb|Kleene|1952|p=83}}.</ref> A theory <math>T</math> is consistent if there is no [[Formula (mathematical logic)|formula]] <math>\varphi</math> such that both <math>\varphi</math> and its negation <math>\lnot\varphi</math> are elements of the set of consequences of <math>T</math>. Let <math>A</math> be a set of [[Closed-form expression|closed sentences]] (informally "axioms") and <math>\langle A\rangle</math> the set of closed sentences provable from <math>A</math> under some (specified, possibly implicitly) formal deductive system. The set of axioms <math>A</math> is '''consistent''' when there is no formula <math>\varphi</math> such that <math>\varphi \in \langle A \rangle</math> and <math> \lnot \varphi \in \langle A \rangle</math>. A ''trivial'' theory (i.e., one which proves every sentence in the language of the theory) is clearly inconsistent. Conversely, in an [[principle of explosion|explosive]] [[formal system]] (e.g., classical or intuitionistic propositional or first-order logics) every inconsistent theory is trivial.<ref name="Carnielli">{{cite book|last1=Carnielli|first1=Walter|last2=Coniglio|first2=Marcelo Esteban|title=Paraconsistent logic: consistency, contradiction and negation|language=en|series=Logic, Epistemology, and the Unity of Science|volume=40|publisher=Springer|location=Cham|date=2016|doi=10.1007/978-3-319-33205-5 |isbn=978-3-319-33203-1|mr=3822731|zbl=1355.03001}}</ref>{{rp|7}} Consistency of a theory is a [[syntactic]] notion, whose [[semantic]] counterpart is [[satisfiable theory|satisfiability]]. A theory is satisfiable if it has a [[Model theory#First-order logic|model]], i.e., there exists an [[interpretation (logic)|interpretation]] under which all [[axiom]]s in the theory are true.<ref>{{cite book |title=A Shorter Model Theory |first=Wilfrid |last=Hodges |page=37 |location=New York |publisher=Cambridge University Press |year=1997 |quote=Let <math>L</math> be a signature, <math>T</math> a theory in <math>L_{\infty \omega}</math> and <math>\varphi</math> a sentence in <math>L_{\infty\omega}</math>. We say that <math>\varphi</math> is a ''consequence'' of <math>T</math>, or that <math>T</math> ''entails'' <math>\varphi</math>, in symbols <math>T \vdash \varphi</math>, if every model of <math>T</math> is a model of <math>\varphi</math>. (In particular if <math>T</math> has no models then <math>T</math> entails <math>\varphi</math>.)<br>'' Warning'': we don't require that if <math>T \vdash \varphi</math> then there is a proof of <math>\varphi</math> from <math>T</math>. In any case, with infinitary languages, it's not always clear what would constitute proof. Some writers use <math>T\vdash\varphi</math> to mean that <math>\varphi</math> is deducible from <math>T</math> in some particular formal proof calculus, and they write <math>T \models \varphi</math> for our notion of entailment (a notation which clashes with our <math>A \models \varphi</math>). For first-order logic, the two kinds of entailment coincide by the completeness theorem for the proof calculus in question.<br>We say that <math>\varphi</math> is ''valid'', or is a ''logical theorem'', in symbols <math>\vdash \varphi</math>, if <math>\varphi</math> is true in every <math>L</math>-structure. We say that <math>\varphi</math> is ''consistent'' if <math>\varphi</math> is true in some <math>L</math>-structure. Likewise, we say that a theory <math>T</math> is ''consistent'' if it has a model.<br> 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 ...)</ref> This is what ''consistent'' meant in traditional [[Term logic|Aristotelian logic]], although in contemporary mathematical logic the term ''[[satisfiable]]'' is used instead.

In a [[soundness|sound formal system]], 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 [[Mathematical logic#Formal logical systems|logic]], the logic is called '''[[Completeness (logic)#Refutation-completeness|complete]]'''.{{citation needed|date=May 2012}} The completeness of the [[propositional calculus]] was proved by [[Paul Bernays]] in 1918{{Citation needed|date=October 2009}}<ref>{{harvnb|van Heijenoort|1967|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''.</ref> and [[Emil Post]] in 1921,<ref>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 {{harvnb|van Heijenoort|1967|pp=264ff}}. Also {{Harvnb|Tarski|1946|pp=134ff}}.</ref> while the completeness of (first order) [[predicate calculus]] was proved by [[Kurt Gödel]] in 1930,<ref>cf van Heijenoort's commentary and Gödel's 1930 ''The completeness of the axioms of the functional calculus of logic'' in {{Harvnb|van Heijenoort|1967|pp=582ff}}.</ref> and consistency proofs for arithmetics restricted with respect to the [[Mathematical induction|induction axiom schema]] were proved by Ackermann (1924), von Neumann (1927) and Herbrand (1931).<ref>cf van Heijenoort's commentary and Herbrand's 1930 ''On the consistency of arithmetic'' in {{Harvnb|van Heijenoort|1967|pp=618ff}}.</ref> Stronger logics, such as [[second-order logic]], are not complete.

A '''consistency proof''' is a [[mathematical proof]] that a particular theory is consistent.<ref>A consistency proof often assumes the consistency of another theory. In most cases, this other theory is  [[Zermelo–Fraenkel set theory]] with or without the [[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).</ref> 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).

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]] (or equivalently the [[Normalization property|normalization]] of the [[Curry-Howard|underlying calculus]] 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==
{{Unreferenced section|date=May 2026}}
In theories of arithmetic, such as [[Peano arithmetic]], there is an intricate relationship between the consistency of the theory and its [[completeness (logic)|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 [[recursively enumerable]] 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 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&mdash;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.

Because consistency of ZF is not provable in ZF, the weaker notion '''{{vanchor|relative consistency}}''' is interesting in set theory (and in other sufficiently expressive axiomatic systems). If ''T'' is a [[theory (mathematical logic)|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 [[Independence (mathematical logic)|independent]] of ''T''.

== First-order logic ==
{{Unreferenced section|date=May 2026}}

===Notation===
In the following context of [[mathematical logic]], the [[turnstile symbol]] <math>\vdash</math> means "provable from". That is, <math>a\vdash b</math> reads: ''b'' is provable from ''a'' (in some specified formal system).

===Definition===
*A set of [[Well-formed formula|formulas]] <math>\Phi</math> in first-order logic is '''consistent''' (written <math>\operatorname{Con} \Phi</math>) if there is no formula <math>\varphi</math> such that <math>\Phi \vdash \varphi</math> and <math>\Phi \vdash \lnot\varphi</math>.  Otherwise <math>\Phi</math> is '''inconsistent''' (written <math>\operatorname{Inc}\Phi</math>).
*<math>\Phi</math> is said to be '''simply consistent''' if for no formula <math>\varphi</math> of <math>\Phi</math>, both <math>\varphi</math> and the [[negation]] of <math>\varphi</math> are theorems of <math>\Phi</math>.{{clarify|reason=Assuming that 'provable from' and 'theorem of' is equivalent, there seems to be no difference between 'consistent' and 'simply consistent'. If that is true, both definitions should be joined into a single one. If not, the difference should be made clear.|date=September 2018}}
*<math>\Phi</math> is said to be '''absolutely consistent''' or '''Post consistent''' if at least one formula in the language of <math>\Phi</math> is not a theorem of <math>\Phi</math>.
*<math>\Phi</math> is said to be '''maximally consistent''' if <math>\Phi</math> is consistent and for every formula <math>\varphi</math>, <math>\operatorname{Con} (\Phi \cup \{\varphi\})</math> implies <math>\varphi \in \Phi</math>.
*<math>\Phi</math> is said to '''contain witnesses''' if for every formula of the form <math>\exists x \,\varphi</math> there exists a [[Term (logic)|term]] <math>t</math> such that <math>(\exists x \, \varphi \to \varphi {t \over x}) \in \Phi</math>, where <math>\varphi {t \over x}</math> denotes the [[substitution (logic)|substitution]] of each <math>x</math> in <math>\varphi</math> by a <math>t</math>; see also [[First-order logic]].{{citation needed|date=September 2018}}

===Basic results===
# The following are equivalent:
## <math>\operatorname{Inc}\Phi</math>
## For all <math>\varphi,\; \Phi \vdash \varphi.</math>
# Every satisfiable set of formulas is consistent, where a set of formulas <math>\Phi</math> is satisfiable if and only if there exists a model <math>\mathfrak{I}</math> such that <math>\mathfrak{I} \vDash \Phi </math>.
# For all <math>\Phi</math> and <math>\varphi</math>:
## if not <math> \Phi \vdash \varphi</math>, then <math>\operatorname{Con}\left( \Phi \cup \{\lnot\varphi\}\right)</math>;
## if <math>\operatorname{Con}\Phi</math> and <math>\Phi \vdash \varphi</math>, then <math> \operatorname{Con} \left(\Phi \cup \{\varphi\}\right)</math>;
## if <math>\operatorname{Con}\Phi</math>, then <math>\operatorname{Con}\left( \Phi \cup \{\varphi\}\right)</math> or <math>\operatorname{Con}\left( \Phi \cup \{\lnot \varphi\}\right)</math>.
# Let <math>\Phi</math> be a maximally consistent set of formulas and suppose it contains [[Witness (mathematics)|witnesses]].  For all <math>\varphi</math> and <math> \psi </math>:
## if <math> \Phi \vdash \varphi</math>, then <math>\varphi \in \Phi</math>,
## either <math>\varphi \in \Phi</math> or <math>\lnot \varphi \in \Phi</math>,
## <math>(\varphi \lor \psi) \in \Phi</math> if and only if <math>\varphi \in \Phi</math> or <math>\psi \in \Phi</math>,
## if <math>(\varphi\to\psi) \in \Phi</math> and <math>\varphi \in \Phi </math>, then <math>\psi \in \Phi</math>,
## <math>\exists x \, \varphi \in \Phi</math> if and only if there is a term <math>t</math> such that <math>\varphi{t \over x}\in\Phi</math>.{{citation needed|date=September 2018}}

===Henkin's theorem===
Let <math>S</math> be a [[signature (logic)|set of symbols]]. Let <math>\Phi</math> be a maximally consistent set of <math>S</math>-formulas containing [[Witness (mathematics)#Henkin witnesses|witnesses]].

Define an [[equivalence relation]] <math>\sim</math> on the set of <math>S</math>-terms by <math>t_0 \sim t_1</math> if <math>\; t_0 \equiv t_1 \in \Phi</math>, where <math>\equiv</math> denotes [[First-order logic#Equality and its axioms|equality]]. Let <math>\overline t</math> denote the [[equivalence class]] of terms containing <math>t </math>; and let <math>T_\Phi := \{ \; \overline t \mid t \in T^S \} </math> where <math>T^S </math> is the set of terms based on the set of symbols <math>S</math>.

Define the <math>S</math>-[[Structure (mathematical logic)|structure]] <math>\mathfrak T_\Phi </math> over <math> T_\Phi </math>, also called the '''term-structure''' corresponding to <math>\Phi</math>, by:

# for each <math>n</math>-ary relation symbol <math>R \in S</math>, define <math>R^{\mathfrak T_\Phi} \overline {t_0} \ldots \overline {t_{n-1}}</math> if <math>\; R t_0 \ldots t_{n-1} \in \Phi;</math><ref>This definition is independent of the choice of <math>t_i</math> due to the substitutivity properties of <math>\equiv</math> and the maximal consistency of <math>\Phi</math>.</ref>
# for each <math>n</math>-ary function symbol <math>f \in S</math>, define <math>f^{\mathfrak T_\Phi} (\overline {t_0} \ldots \overline {t_{n-1}}) := \overline {f t_0 \ldots t_{n-1}};</math>
# for each constant symbol <math>c \in S</math>, define <math>c^{\mathfrak T_\Phi}:= \overline c.</math>

Define a variable assignment <math>\beta_\Phi</math> by <math>\beta_\Phi (x) := \bar x</math> for each variable <math>x</math>. Let <math>\mathfrak I_\Phi := (\mathfrak T_\Phi,\beta_\Phi)</math> be the '''term [[Interpretation (logic)#First-order logic|interpretation]]''' associated with <math>\Phi</math>.

Then for each <math>S</math>-formula <math>\varphi</math>:

{{center|1=
<math>\mathfrak I_\Phi \vDash \varphi</math> if and only if <math> \; \varphi \in \Phi.</math>{{citation needed|date=September 2018}}
}}

===Sketch of proof===
There are several things to verify. First, that <math>\sim</math> 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 <math>\sim</math> is an equivalence relation and also requires a proof that (1) and (2) are independent of the choice of <math> t_0, \ldots ,t_{n-1} </math> class representatives.  Finally, <math> \mathfrak I_\Phi \vDash \varphi </math> can be verified by induction on formulas.

== Model theory ==

In [[Zermelo–Fraenkel set theory|ZFC set theory]] with classical [[first-order logic]],<ref>the common case in many applications to other areas of mathematics as well as the ordinary mode of reasoning of [[informal mathematics]] in calculus and applications to physics, chemistry, engineering</ref> an '''inconsistent''' theory <math>T</math> is one such that there exists a closed sentence <math>\varphi</math> such that <math>T</math> contains both <math>\varphi</math> and its negation <math>\varphi'</math>. A '''consistent''' theory is one such that the following [[Logical equivalence|logically equivalent]] conditions hold

#<math>\{\varphi,\varphi'\}\not\subseteq T</math><ref>according to [[De Morgan's laws]]</ref>
#<math>\varphi'\not\in T \lor \varphi\not\in T</math>

==See also==
{{Portal|Philosophy}}
{{Wikiquote}}
*{{annotated link|Cognitive dissonance}}
*{{annotated link|Equiconsistency}}
*{{annotated link|Hilbert's problems}}
*{{annotated link|Hilbert's second problem}}
*{{annotated link|Jan Łukasiewicz}}
*{{annotated link|Paraconsistent logic}}
*{{annotated link|ω-consistency}}
*{{annotated link|Gentzen's consistency proof}}
*{{annotated link|Proof by contradiction}}
*{{annotated link|Self-refuting idea}}

==Notes==
{{Reflist}}

==References==
* {{cite journal |last1=Gödel |first1=Kurt |title=Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I |journal=Monatshefte für Mathematik und Physik |date=1 December 1931 |volume=38 |issue=1 |pages=173–198 |doi=10.1007/BF01700692}}
*{{cite book |first=Stephen |last=Kleene |author-link=Stephen Kleene |year=1952 |title=Introduction to Metamathematics |publisher=North-Holland |location=New York |isbn=0-7204-2103-9 }} 10th impression 1991.
*{{cite book |first=Hans |last=Reichenbach |author-link=Hans Reichenbach |year=1947 |title=Elements of Symbolic Logic |publisher=Dover |location=New York |isbn=0-486-24004-5 }}
*{{cite book |first=Alfred |last=Tarski |author-link=Alfred Tarski |year=1946 |title=Introduction to Logic and to the Methodology of Deductive Sciences |edition=Second |publisher=Dover |location=New York |isbn=0-486-28462-X }}
*{{cite book |first=Jean |last=van Heijenoort |author-link=Jean van Heijenoort |year=1967 |title=From Frege to Gödel: A Source Book in Mathematical Logic |publisher=Harvard University Press |location=Cambridge, MA |isbn=0-674-32449-8 }} (pbk.)
*{{cite book |title=[[The Cambridge Dictionary of Philosophy]] |chapter=Consistency }}
*{{cite book |first1=H. D. |last1=Ebbinghaus |first2=J. |last2=Flum |first3=W. |last3=Thomas |title=Mathematical Logic }}
*{{cite book |last=Jevons |first=W. S. |year=1870 |title=Elementary Lessons in Logic }}

==External links==
{{wiktionary}}
*{{cite encyclopedia |first=Chris |last=Mortensen |url=http://plato.stanford.edu/entries/mathematics-inconsistent/ |title=Inconsistent Mathematics |encyclopedia=[[Stanford Encyclopedia of Philosophy]] |year=2017 }}

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