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{{Short description|Non-contradiction of a theory}}{{Other uses}}In classical deductive logic, a consistent theory is one that does not lead to a logical contradiction.{{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}}. 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, i.e., there exists an interpretation under which all formulas in the theory are true. This is the sense used in traditional Aristotelian logic, although in contemporary mathematical logic the term satisfiable is used instead. The syntactic definition states a theory T is consistent if there is no formula varphi such that both varphi and its negation lnotvarphi are elements of the set of consequences of T. Let A be a set of closed sentences (informally “axioms“) and langle Arangle the set of closed sentences provable from A under some (specified, possibly implicitly) formal deductive system. The set of axioms A is consistent when there is no formula varphi such that varphi in langle A rangle and lnot varphi in langle A rangle.BOOK, A Shorter Model Theory, Wilfrid, Hodges, 37, New York, Cambridge University Press, 1997, Let L be a signature, T a theory in L_{infty omega} and varphi a sentence in L_{inftyomega}. We say that varphi is a consequence of T, or that T entails varphi, in symbols T vdash varphi, if every model of T is a model of varphi. (In particular if T has no models then T entails varphi.) Warning: we don’t require that if T vdash varphi then there is a proof of varphi from T. In any case, with infinitary languages, it’s not always clear what would constitute proof. Some writers use Tvdashvarphi to mean that varphi is deducible from T in some particular formal proof calculus, and they write T models varphi for our notion of entailment (a notation which clashes with our 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 varphi is valid, or is a logical theorem, in symbols vdash varphi, if varphi is true in every L-structure. We say that varphi is consistent if varphi is true in some L-structure. Likewise, we say that a theory 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 ...)If there exists a deductive system for which these semantic and syntactic definitions are equivalent for any theory formulated in a particular deductive logic, the logic is called complete.{{citation needed|date=May 2012}} The completeness of the sentential calculus was proved by Paul Bernays in 1918{{Citation needed|date=October 2009}}{{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. and Emil Post in 1921,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}}. while the completeness of predicate calculus was proved by Kurt Gödel in 1930,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}}. and consistency proofs for arithmetics restricted with respect to the induction axiom schema were proved by Ackermann (1924), von Neumann (1927) and Herbrand (1931).cf van Heijenoort’s commentary and Herbrand’s 1930 On the consistency of arithmetic in {{Harvnb|van Heijenoort|1967|pp=618ff}}. Stronger logics, such as second-order logic, are not complete.A consistency proof is a mathematical proof that a particular theory is consistent.Informally, Zermelo–Fraenkel set theory is ordinarily assumed; some dialects of informal mathematics customarily assume the axiom of choice in addition. 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 of the 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

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 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—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 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 thatif 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.

First-order logic

Notation

In the following context of mathematical logic, the turnstile symbol vdash means “provable from”. That is, avdash b reads: b is provable from a (in some specified formal system).

Definition

• A set of formulas Phi in first-order logic is consistent (written operatorname{Con} Phi) if there is no formula varphi such that Phi vdash varphi and Phi vdash lnotvarphi. Otherwise Phi is inconsistent (written operatorname{Inc}Phi).
• Phi is said to be simply consistent if for no formula varphi of Phi, both varphi and the negation of varphi are theorems of Phi.{{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}}
• Phi is said to be absolutely consistent or Post consistent if at least one formula in the language of Phi is not a theorem of Phi.
• Phi is said to be maximally consistent if Phi is consistent and for every formula varphi, operatorname{Con} (Phi cup {varphi}) implies varphi in Phi.
• Phi is said to contain witnesses if for every formula of the form exists x ,varphi there exists a term t such that (exists x , varphi to varphi {t over x}) in Phi, where varphi {t over x} denotes the substitution of each x in varphi by a t; see also First-order logic.{{citation needed|date=September 2018}}

Basic results

1. The following are equivalent:
   • operatorname{Inc}Phi
   • For all varphi,; Phi vdash varphi.
2. Every satisfiable set of formulas is consistent, where a set of formulas Phi is satisfiable if and only if there exists a model mathfrak{I} such that mathfrak{I} vDash Phi .
3. For all Phi and varphi:
   • if not Phi vdash varphi, then operatorname{Con}left( Phi cup {lnotvarphi}right);
   • if operatorname{Con}Phi and Phi vdash varphi, then operatorname{Con} left(Phi cup {varphi}right);
   • if operatorname{Con}Phi, then operatorname{Con}left( Phi cup {varphi}right) or operatorname{Con}left( Phi cup {lnot varphi}right).
4. Let Phi be a maximally consistent set of formulas and suppose it contains witnesses. For all varphi and psi :
   • if Phi vdash varphi, then varphi in Phi,
   • either varphi in Phi or lnot varphi in Phi,
   • (varphi lor psi) in Phi if and only if varphi in Phi or psi in Phi,
   • if (varphitopsi) in Phi and varphi in Phi , then psi in Phi,
   • exists x , varphi in Phi if and only if there is a term t such that varphi{t over x}inPhi.{{citation needed|date=September 2018}}

Henkin’s theorem

Let S be a set of symbols. Let Phi be a maximally consistent set of S-formulas containing witnesses.Define an equivalence relation sim on the set of S-terms by t_0 sim t_1 if ; t_0 equiv t_1 in Phi, where equiv denotes equality. Let overline t denote the equivalence class of terms containing t ; and let T_Phi := { ; overline t mid t in T^S } where T^S is the set of terms based on the set of symbols S.Define the S-structure mathfrak T_Phi over T_Phi , also called the term-structure corresponding to Phi, by:

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

Define a variable assignment beta_Phi by beta_Phi (x) := bar x for each variable x. Let mathfrak I_Phi := (mathfrak T_Phi,beta_Phi) be the term interpretation associated with Phi.Then for each S-formula varphi:{{center|1=mathfrak I_Phi vDash varphi if and only if ; varphi in Phi.{{citation needed|date=September 2018}}}}

Sketch of proof

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

Model theory

In ZFC set theory with classical first-order logic,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 an inconsistent theory T is one such that there exists a closed sentence varphi such that T contains both varphi and its negation varphi’. A consistent theory is one such that the following logically equivalent conditions hold

1. {varphi,varphi’}notsubseteq Taccording to De Morgan’s laws
2. varphi’notin T lor varphinotin T

See also

• Cognitive dissonance
• Equiconsistency
• Hilbert’s problems
• Hilbert’s second problem
• Jan Łukasiewicz
• Paraconsistent logic
• ω-consistency
• Gentzen’s consistency proof
• Proof by contradiction

Footnotes

{{Reflist}}

References

• JOURNAL, Gödel, Kurt, Ãœber formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I, Monatshefte für Mathematik und Physik, 1 December 1931, 38, 1, 173–198, 10.1007/BF01700692,
• BOOK, Stephen, Kleene, Stephen Kleene, 1952, Introduction to Metamathematics, North-Holland, New York, 0-7204-2103-9, 10th impression 1991.
• BOOK, Hans, Reichenbach, Hans Reichenbach, 1947, Elements of Symbolic Logic, Dover, New York, 0-486-24004-5,
• BOOK, Alfred, Tarski, Alfred Tarski, 1946, Introduction to Logic and to the Methodology of Deductive Sciences, Second, Dover, New York, 0-486-28462-X,
• BOOK, Jean, van Heijenoort, Jean van Heijenoort, 1967, From Frege to Gödel: A Source Book in Mathematical Logic, Harvard University Press, Cambridge, MA, 0-674-32449-8, (pbk.)
• BOOK, The Cambridge Dictionary of Philosophy, Consistency,
• BOOK, H. D., Ebbinghaus, J., Flum, W., Thomas, Mathematical Logic,
• BOOK, Jevons, W. S., 1870, Elementary Lessons in Logic,

External links

{{wiktionary}}

• ENCYCLOPEDIA, Chris, Mortensen,plato.stanford.edu/entries/mathematics-inconsistent/, Inconsistent Mathematics, Stanford Encyclopedia of Philosophy, 2017,

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