Consistency Explained

In classical deductive logic, a consistent theory is one that does not lead to a logical contradiction.^[1] A theory

T is consistent if there is no formula \varphi such that both \varphi and its negation ln ot\varphi are elements of the set of consequences of T . Let A be a set of closed sentences (informally "axioms") and \langle A\rangle 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 ln ot \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 formal system (e.g., classical or intuitionistic propositional or first-order logics) every inconsistent theory is trivial.^[2] Consistency of a theory is a syntactic notion, whose semantic counterpart is satisfiability. A theory is satisfiable if it has a model, i.e., there exists an interpretation under which all axioms in the theory are true.^[3] This is what consistent meant in traditional Aristotelian logic, although in contemporary mathematical logic the term satisfiable is used instead.

In a 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 logic, the logic is called complete. The completeness of the propositional calculus was proved by Paul Bernays in 1918^[4] and Emil Post in 1921,^[5] while the completeness of (first order) predicate calculus was proved by Kurt Gödel in 1930,^[6] and consistency proofs for arithmetics restricted with respect to the induction axiom schema were proved by Ackermann (1924), von Neumann (1927) and Herbrand (1931).^[7] Stronger logics, such as second-order logic, are not complete.

A consistency proof is a mathematical proof that a particular theory is consistent.^[8] 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 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

\vdash means "provable from". That is, a\vdash b reads: b is provable from a (in some specified formal system).

Definition

\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 ln ot\varphi . 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 .\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.

Basic results

1. The following are equivalent:

\operatorname{Inc}\Phi

1. 1. For all

\varphi, \Phi \vdash \varphi.

1. Every satisfiable set of formulas is consistent, where a set of formulas

\Phi is satisfiable if and only if there exists a model ak{I} such that ak{I} \vDash \Phi .

1. For all

\Phi and \varphi :

1. 1. if not

\Phi \vdash \varphi , then \operatorname{Con}\left( \Phi \cup \{ ln ot\varphi\}\right) ;

1. 1. if

\operatorname{Con}\Phi and \Phi \vdash \varphi , then \operatorname{Con} \left(\Phi \cup \{\varphi\}\right) ;

1. 1. if

\operatorname{Con}\Phi , then \operatorname{Con}\left( \Phi \cup \{\varphi\}\right) or \operatorname{Con}\left( \Phi \cup \{ ln ot \varphi\}\right) .

1. Let

\Phi be a maximally consistent set of formulas and suppose it contains witnesses. For all \varphi and \psi :

1. 1. if

\Phi \vdash \varphi , then \varphi \in \Phi ,

1. 1. either

\varphi \in \Phi or ln ot \varphi \in \Phi ,(\varphi \lor \psi) \in \Phi if and only if \varphi \in \Phi or \psi \in \Phi ,

1. 1. if

(\varphi\to\psi) \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}\in\Phi .

Henkin's theorem

Let

S be a set of symbols. Let \Phi be a maximally consistent set of S -formulas containing witnesses.\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 ak 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 ^{ak T _{\Phi}} \overline {t _{0}} \ldots \overline {t _{n-1} } if R t _{0} \ldots t _{n-1} \in \Phi; ^[9]

1. for each

n -ary function symbol f \in S , define f ^{ak T _{\Phi}} (\overline {t _{0}} \ldots \overline {t _{n-1} }) := \overline ;

1. for each constant symbol

c \in S , define c ^{ak T _{\Phi}} : = \overline c.

Define a variable assignment

\beta _{\Phi} by \beta _{\Phi} (x) : = \bar x for each variable x . Let ak I _{\Phi} : = ( ak T _{\Phi,\beta} _{\Phi)} be the term interpretation associated with \Phi .

Then for each

S -formula \varphi :

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, ak I _{\Phi} \vDash \varphi can be verified by induction on formulas.

Model theory

In ZFC set theory with classical first-order logic,^[10] 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\{\varphi,\varphi'\}\not\subseteq T ^[11] \varphi'\not\in T \lor \varphi\not\in 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

References

• Gödel . Kurt . Über 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: Kleene, Stephen . Stephen Kleene . 1952 . Introduction to Metamathematics . North-Holland . New York . 0-7204-2103-9 . 10th impression 1991.
• Book: Reichenbach, Hans . Hans Reichenbach . 1947 . Elements of Symbolic Logic . Dover . New York . 0-486-24004-5 .
• Book: Tarski, Alfred . Alfred Tarski . 1946 . Introduction to Logic and to the Methodology of Deductive Sciences . Second . Dover . New York . 0-486-28462-X .
• Book: van Heijenoort, Jean . 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: . Consistency .
• Book: H. D. . Ebbinghaus . J. . Flum . W. . Thomas . Mathematical Logic .
• Book: Jevons, W. S. . 1870 . Elementary Lessons in Logic .

External links

• Encyclopedia: Chris . Mortensen . Inconsistent Mathematics . . 2017 .

Notes and References

1. 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". 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, . 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 –
2. Book: Carnielli. Walter. Coniglio. Marcelo Esteban. Paraconsistent logic: consistency, contradiction and negation. en. Logic, Epistemology, and the Unity of Science. 40. Springer. Cham. 2016. 10.1007/978-3-319-33205-5 . 978-3-319-33203-1. 3822731. 1355.03001.
3. Book: Hodges, Wilfrid . A Shorter Model Theory . 37 . New York . Cambridge University Press . 1997 . Let L be a signature, T a theory in L _{infty} and \varphi a sentence in L _{infty\omega} . 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 T\vdash\varphi 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 ...)
4. 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.
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 . Also .
6. cf van Heijenoort's commentary and Gödel's 1930 The completeness of the axioms of the functional calculus of logic in .
7. cf van Heijenoort's commentary and Herbrand's 1930 On the consistency of arithmetic in .
8. 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).
9. This definition is independent of the choice of t _{i} due to the substitutivity properties of \equiv and the maximal consistency of \Phi .
10. 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
11. according to De Morgan's laws