Consistency

In classical deductive logic, a consistent theory is one that does not contain a contradiction.^[1]^[2] The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent if and only 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 and only if there is no formula $\phi$ such that both $\phi$ and its negation $\lnot \phi$ are elements of the set $T$ . 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 $\langle A\rangle$ is.^[3]

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 sentential calculus was proved by Paul Bernays in 1918^[4] and Emil Post in 1921,^[5] while the completeness of 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 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.

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. 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 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 independent of T.

First-order logic

Notation

$\vdash$ (Turnstile symbol) in the following context of Mathematical logic, means "provable from". That is, a $\vdash$ b reads: b is provable from a (in some specified formal system). See List of logic symbols. In other cases, the turnstile symbol may mean implies; permits the derivation of. See: List of mathematical symbols.

Definition

A set of formulas $\Phi$ in first-order logic is consistent (written Con $\Phi$ ) if and only if there is no formula $\phi$ such that $\Phi \vdash \phi$ and $\Phi \vdash \lnot\phi$ . Otherwise $\Phi$ is inconsistent and is written Inc $\Phi$ .

$\Phi$ is said to be simply consistent if and only if for no formula $\phi$ of $\Phi$ , both $\phi$ and the negation of $\phi$ are theorems of $\Phi$ .

$\Phi$ is said to be absolutely consistent or Post consistent if and only if at least one formula of $\Phi$ is not a theorem of $\Phi$ .

$\Phi$ is said to be maximally consistent if and only if for every formula $\phi$ , if Con ( $\Phi \cup \phi$ ) then $\phi \in \Phi$ .

$\Phi$ is said to contain witnesses if and only if for every formula of the form $\exists x \phi$ there exists a term $t$ such that $(\exists x \phi \to \phi {t \over x}) \in \Phi$ . See First-order logic.

Basic results

The following are equivalent:
1. Inc $\Phi$
2. For all $\phi,\; \Phi \vdash \phi.$
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 $\mathfrak{I} \vDash \Phi$ .
For all $\Phi$ and $\phi$ :
1. if not $\Phi \vdash \phi$ , then Con $\left( \Phi \cup \{\lnot\phi\}\right)$ ;
2. if Con $\Phi$ and $\Phi \vdash \phi$ , then Con $\left(\Phi \cup \{\phi\}\right)$ ;
3. if Con $\Phi$ , then Con $\left( \Phi \cup \{\phi\}\right)$ or Con $\left( \Phi \cup \{\lnot \phi\}\right)$ .
Let $\Phi$ be a maximally consistent set of formulas and contain witnesses. For all $\phi$ and $\psi$ :
1. if $\Phi \vdash \phi$ , then $\phi \in \Phi$ ,
2. either $\phi \in \Phi$ or $\lnot \phi \in \Phi$ ,
3. $(\phi \or \psi) \in \Phi$ if and only if $\phi \in \Phi$ or $\psi \in \Phi$ ,
4. if $(\phi\to\psi) \in \Phi$ and $\phi \in \Phi$ , then $\psi \in \Phi$ ,
5. $\exists x \phi \in \Phi$ if and only if there is a term $t$ such that $\phi{t \over x}\in\Phi$ .

Henkin's theorem

Let $\Phi$ be a maximally consistent set of $S$ -formulas containing witnesses.

Define a binary relation $\sim$ on the set of $S$ -terms such that $t_0 \sim t_1$ if and only if $\; t_0 \equiv t_1 \in \Phi$ ; and let ${\overline {t}}$ denote the equivalence class of terms containing $t$ ; and let $T_{\Phi} := \{ \; \overline t \; |\; t \in T^S \}$ where $T^{S}$ is the set of terms based on the symbol set $S$ .

Define the $S$ -structure $\mathfrak T_{\Phi}$ over $T_{\Phi }$ the term-structure corresponding to $\Phi$ by:

for $n$ -ary $R \in S$ , $R^{\mathfrak T_{\Phi}} \overline {t_0} \ldots \overline {t_{n-1}}$ if and only if $\; R t_0 \ldots t_{n-1} \in \Phi$ ;
for $n$ -ary $f \in S$ , $f^{\mathfrak T_{\Phi}} (\overline {t_0} \ldots \overline {t_{n-1}}) := \overline {f t_0 \ldots t_{n-1}}$ ;
for $c \in S$ , $c^{\mathfrak T_{\Phi}}:= \overline c$ .

Let $\mathfrak I_{\Phi} := (\mathfrak T_{\Phi},\beta_{\Phi})$ be the term interpretation associated with $\Phi$ , where $\beta _{\Phi} (x) := \bar x$ .

For all $\phi$ , $\; \mathfrak I_{\Phi} \vDash \phi$ if and only if $\; \phi \in \Phi$ .

Sketch of proof

There are several things to verify. First, that $\sim$ is 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, $\mathfrak I_{\Phi} \vDash \Phi$ can be verified by induction on formulas.

Model theory

In ZFC set theory with classical first-order logic,^[9] an inconsistent theory $T$ is one such that there exists a closed sentence $\phi$ such that $T$ contains both $\phi$ and its negation $\phi '$ . A consistent theory is one such that the following logically equivalent conditions hold

$\{\phi ,\phi '\}\not \subseteq T$ ^[10]
$\phi '\not \in T\lor \phi \not \in T$

Footnotes

↑ 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". Gödel in his 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 Gödel 1931 van Heijenoort 1967:601. 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 1947:68]" cf Tarski 1946:3. 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 Kleene 1952:83.
↑ see Paraconsistent logic
↑
Let $L$ be a signature, $T$ a theory in $L_{\infty \omega }$ and $\phi$ a sentence in $L_{{\infty \omega }}$ . We say that $\phi$ is a consequence of $T$ , or that $T$ entails $\phi$ , in symbols $T\vdash \phi$ , if every model of $T$ is a model of $\phi$ . (In particular if $T$ has no models then $T$ entails $\phi$ .) Warning: we don't require that if $T\vdash \phi$ then there is a proof of $\phi$ from $T$ . In any case, with infinitary languages it's not always clear what would constitute a proof. Some writers use $T\vdash \phi$ to mean that $\phi$ is deducible from $T$ in some particular formal proof calculus, and they write $T\models \phi$ for our notion of entailment (a notation which clashes with our $A\models \phi$ ). For first-order logic the two kinds of entailment coincide by the completeness theorem for the proof calculus in question. We say that $\phi$ is valid, or is a logical theorem, in symbols $\vdash \phi$ , if $\phi$ is true in every $L$ -structure. We say that $\phi$ is consistent if $\phi$ 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 definition of Mod(T) on p. 30 ...)
A Shorter Model Theory by Wilfrid Hodges, p. 37
↑ van Heijenoort 1967:265 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.
↑ 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 1967:264ff. Also Tarski 1946:134ff.
↑ cf van Heijenoort's commentary and Gödel's 1930 The completeness of the axioms of the functional calculus of logic in van Heijenoort 1967:582ff
↑ cf van Heijenoort's commentary and Herbrand's 1930 On the consistency of arithmetic in van Heijenoort 1967:618ff.
↑ Informally, Zermelo–Fraenkel set theory is ordinarily assumed; some dialects of informal mathematics customarily assume the axiom of choice in addition.
↑ 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
↑ according to De Morgan's laws

References

Stephen Kleene, 1952 10th impression 1991, Introduction to Metamathematics, North-Holland Publishing Company, Amsterday, New York, ISBN 0-7204-2103-9.
Hans Reichenbach, 1947, Elements of Symbolic Logic, Dover Publications, Inc. New York, ISBN 0-486-24004-5,
Alfred Tarski, 1946, Introduction to Logic and to the Methodology of Deductive Sciences, Second Edition, Dover Publications, Inc., New York, ISBN 0-486-28462-X.
Jean van Heijenoort, 1967, From Frege to Gödel: A Source Book in Mathematical Logic, Harvard University Press, Cambridge, MA, ISBN 0-674-32449-8 (pbk.)
The Cambridge Dictionary of Philosophy, consistency
H.D. Ebbinghaus, J. Flum, W. Thomas, Mathematical Logic
Jevons, W.S., 1870, Elementary Lessons in Logic

External links

Chris Mortensen, Inconsistent Mathematics, Stanford Encyclopedia of Philosophy

‌Logical truth ⊤
Functional:	truth value truth function ⊨ tautology
Formal:	theory formal proof theorem
Negation	⊥ false contradiction inconsistency

Metalogic Metamathematics
Cantor's theorem Entscheidungsproblem Church–Turing thesis Consistency Effective method Foundations of mathematics (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

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