In classical deductive logic, a consistent theory is one that does not entail a contradiction. 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 $\backslash varphi$ such that both $\backslash varphi$ and its negation $\backslash lnot\backslash varphi$ are elements of the set of consequences of $T$. Let $A$ be a set of closed sentences (informally "axioms") and $\backslash langle\; A\backslash rangle$ the set of closed sentences provable from $A$ under some (specified, possibly implicitly) formal deductive system. The set of axioms $A$ is consistent when $\backslash varphi,\; \backslash lnot\; \backslash varphi\; \backslash in\; \backslash langle\; A\; \backslash rangle$ for no formula $\backslash varphi$.
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 and Emil Post in 1921, while the completeness of predicate calculus was proved by Kurt Gödel in 1930, and consistency proofs for arithmetics restricted with respect to the induction axiom schema were proved by Ackermann (1924), von Neumann (1927) and Herbrand (1931). Stronger logics, such as second-order logic, are not complete.
A consistency proof is a mathematical proof that a particular theory is consistent. 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 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 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 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

$\backslash vdash$ (Turnstile symbol) in the following context of mathematical logic, means "provable from". That is, $a\backslash vdash\; b$ 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 $\backslash Phi$ in first-order logic is consistent (written $\backslash operatorname\; \backslash Phi$) if there is no formula $\backslash varphi$ such that $\backslash Phi\; \backslash vdash\; \backslash varphi$ and $\backslash Phi\; \backslash vdash\; \backslash lnot\backslash varphi$. Otherwise $\backslash Phi$ is inconsistent (written $\backslash operatorname\backslash Phi$). *$\backslash Phi$ is said to be simply consistent if for no formula $\backslash varphi$ of $\backslash Phi$, both $\backslash varphi$ and the negation of $\backslash varphi$ are theorems of $\backslash Phi$. *$\backslash Phi$ is said to be absolutely consistent or Post consistent if at least one formula in the language of $\backslash Phi$ is not a theorem of $\backslash Phi$. *$\backslash Phi$ is said to be maximally consistent if for every formula $\backslash varphi$, if $\backslash operatorname\; (\backslash Phi\; \backslash cup\; \backslash )$ implies $\backslash varphi\; \backslash in\; \backslash Phi$. *$\backslash Phi$ is said to contain witnesses if for every formula of the form $\backslash exists\; x\; \backslash ,\backslash varphi$ there exists a term $t$ such that $(\backslash exists\; x\; \backslash ,\; \backslash varphi\; \backslash to\; \backslash varphi\; )\; \backslash in\; \backslash Phi$, where $\backslash varphi$ denotes the substitution of each $x$ in $\backslash varphi$ by a $t$; see also First-order logic.

Basic results

# The following are equivalent: ## $\backslash operatorname\backslash Phi$ ## For all $\backslash varphi,\backslash ;\; \backslash Phi\; \backslash vdash\; \backslash varphi.$ # Every satisfiable set of formulas is consistent, where a set of formulas $\backslash Phi$ is satisfiable if and only if there exists a model $\backslash mathfrak$ such that $\backslash mathfrak\; \backslash vDash\; \backslash Phi$. # For all $\backslash Phi$ and $\backslash varphi$: ## if not $\backslash Phi\; \backslash vdash\; \backslash varphi$, then $\backslash operatorname\backslash left(\; \backslash Phi\; \backslash cup\; \backslash \backslash right)$; ## if $\backslash operatorname\backslash Phi$ and $\backslash Phi\; \backslash vdash\; \backslash varphi$, then $\backslash operatorname\; \backslash left(\backslash Phi\; \backslash cup\; \backslash \backslash right)$; ## if $\backslash operatorname\backslash Phi$, then $\backslash operatorname\backslash left(\; \backslash Phi\; \backslash cup\; \backslash \backslash right)$ or $\backslash operatorname\backslash left(\; \backslash Phi\; \backslash cup\; \backslash \backslash right)$. # Let $\backslash Phi$ be a maximally consistent set of formulas and suppose it contains witnesses. For all $\backslash varphi$ and $\backslash psi$: ## if $\backslash Phi\; \backslash vdash\; \backslash varphi$, then $\backslash varphi\; \backslash in\; \backslash Phi$, ## either $\backslash varphi\; \backslash in\; \backslash Phi$ or $\backslash lnot\; \backslash varphi\; \backslash in\; \backslash Phi$, ## $(\backslash varphi\; \backslash lor\; \backslash psi)\; \backslash in\; \backslash Phi$ if and only if $\backslash varphi\; \backslash in\; \backslash Phi$ or $\backslash psi\; \backslash in\; \backslash Phi$, ## if $(\backslash varphi\backslash to\backslash psi)\; \backslash in\; \backslash Phi$ and $\backslash varphi\; \backslash in\; \backslash Phi$, then $\backslash psi\; \backslash in\; \backslash Phi$, ## $\backslash exists\; x\; \backslash ,\; \backslash varphi\; \backslash in\; \backslash Phi$ if and only if there is a term $t$ such that $\backslash varphi\backslash in\backslash Phi$.

Henkin's theorem

Let $S$ be a set of symbols. Let $\backslash Phi$ be a maximally consistent set of $S$-formulas containing witnesses. Define an equivalence relation $\backslash sim$ on the set of $S$-terms by $t\_0\; \backslash sim\; t\_1$ if $\backslash ;\; t\_0\; \backslash equiv\; t\_1\; \backslash in\; \backslash Phi$, where $\backslash equiv$ denotes equality. Let $\backslash overline\; t$ denote the equivalence class of terms containing $t$; and let $T\_\backslash Phi\; :=\; \backslash $ where $T^S$ is the set of terms based on the set of symbols $S$. Define the $S$-structure $\backslash mathfrak\; T\_\backslash Phi$ over $T\_\backslash Phi$, also called the term-structure corresponding to $\backslash Phi$, by: # for each $n$-ary relation symbol $R\; \backslash in\; S$, define $R^\; \backslash overline\; \backslash ldots\; \backslash overline$ if $\backslash ;\; R\; t\_0\; \backslash ldots\; t\_\; \backslash in\; \backslash Phi;$ # for each $n$-ary function symbol $f\; \backslash in\; S$, define $f^\; (\backslash overline\; \backslash ldots\; \backslash overline\; )\; :=\; \backslash overline\; ;$ # for each constant symbol $c\; \backslash in\; S$, define $c^:=\; \backslash overline\; c.$ Define a variable assignment $\backslash beta\_\backslash Phi$ by $\backslash beta\_\backslash Phi\; (x)\; :=\; \backslash bar\; x$ for each variable $x$. Let $\backslash mathfrak\; I\_\backslash Phi\; :=\; (\backslash mathfrak\; T\_\backslash Phi,\backslash beta\_\backslash Phi)$ be the term interpretation associated with $\backslash Phi$. Then for each $S$-formula $\backslash varphi$:
$\backslash mathfrak\; I\_\backslash Phi\; \backslash vDash\; \backslash varphi$ if and only if $\backslash ;\; \backslash varphi\; \backslash in\; \backslash Phi.$

Sketch of proof

There are several things to verify. First, that $\backslash sim$ 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 $\backslash sim$ is an equivalence relation and also requires a proof that (1) and (2) are independent of the choice of $t\_0,\; \backslash ldots\; ,t\_$ class representatives. Finally, $\backslash mathfrak\; I\_\backslash Phi\; \backslash vDash\; \backslash varphi$ can be verified by induction on formulas.

** Model theory **

In ZFC set theory with classical first-order logic, an inconsistent theory $T$ is one such that there exists a closed sentence $\backslash varphi$ such that $T$ contains both $\backslash varphi$ and its negation $\backslash varphi\text{'}$. A consistent theory is one such that the following logically equivalent conditions hold
#$\backslash \backslash not\backslash subseteq\; T$according to De Morgan's laws
#$\backslash varphi\text{'}\backslash not\backslash in\; T\; \backslash lor\; \backslash varphi\backslash not\backslash in\; T$

See also

*Equiconsistency *Hilbert's problems *Hilbert's second problem *Jan Łukasiewicz *Paraconsistent logic *ω-consistency *Gentzen's consistency proof *Proof by contradiction

Footnotes

References

* 10th impression 1991. * * * (pbk.) * * *

External links

* {{Authority control Category:Proof theory Category:Hilbert's problems Category:Metalogic

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 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''.

Notation

$\backslash vdash$ (Turnstile symbol) in the following context of mathematical logic, means "provable from". That is, $a\backslash vdash\; b$ 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 $\backslash Phi$ in first-order logic is consistent (written $\backslash operatorname\; \backslash Phi$) if there is no formula $\backslash varphi$ such that $\backslash Phi\; \backslash vdash\; \backslash varphi$ and $\backslash Phi\; \backslash vdash\; \backslash lnot\backslash varphi$. Otherwise $\backslash Phi$ is inconsistent (written $\backslash operatorname\backslash Phi$). *$\backslash Phi$ is said to be simply consistent if for no formula $\backslash varphi$ of $\backslash Phi$, both $\backslash varphi$ and the negation of $\backslash varphi$ are theorems of $\backslash Phi$. *$\backslash Phi$ is said to be absolutely consistent or Post consistent if at least one formula in the language of $\backslash Phi$ is not a theorem of $\backslash Phi$. *$\backslash Phi$ is said to be maximally consistent if for every formula $\backslash varphi$, if $\backslash operatorname\; (\backslash Phi\; \backslash cup\; \backslash )$ implies $\backslash varphi\; \backslash in\; \backslash Phi$. *$\backslash Phi$ is said to contain witnesses if for every formula of the form $\backslash exists\; x\; \backslash ,\backslash varphi$ there exists a term $t$ such that $(\backslash exists\; x\; \backslash ,\; \backslash varphi\; \backslash to\; \backslash varphi\; )\; \backslash in\; \backslash Phi$, where $\backslash varphi$ denotes the substitution of each $x$ in $\backslash varphi$ by a $t$; see also First-order logic.

Basic results

# The following are equivalent: ## $\backslash operatorname\backslash Phi$ ## For all $\backslash varphi,\backslash ;\; \backslash Phi\; \backslash vdash\; \backslash varphi.$ # Every satisfiable set of formulas is consistent, where a set of formulas $\backslash Phi$ is satisfiable if and only if there exists a model $\backslash mathfrak$ such that $\backslash mathfrak\; \backslash vDash\; \backslash Phi$. # For all $\backslash Phi$ and $\backslash varphi$: ## if not $\backslash Phi\; \backslash vdash\; \backslash varphi$, then $\backslash operatorname\backslash left(\; \backslash Phi\; \backslash cup\; \backslash \backslash right)$; ## if $\backslash operatorname\backslash Phi$ and $\backslash Phi\; \backslash vdash\; \backslash varphi$, then $\backslash operatorname\; \backslash left(\backslash Phi\; \backslash cup\; \backslash \backslash right)$; ## if $\backslash operatorname\backslash Phi$, then $\backslash operatorname\backslash left(\; \backslash Phi\; \backslash cup\; \backslash \backslash right)$ or $\backslash operatorname\backslash left(\; \backslash Phi\; \backslash cup\; \backslash \backslash right)$. # Let $\backslash Phi$ be a maximally consistent set of formulas and suppose it contains witnesses. For all $\backslash varphi$ and $\backslash psi$: ## if $\backslash Phi\; \backslash vdash\; \backslash varphi$, then $\backslash varphi\; \backslash in\; \backslash Phi$, ## either $\backslash varphi\; \backslash in\; \backslash Phi$ or $\backslash lnot\; \backslash varphi\; \backslash in\; \backslash Phi$, ## $(\backslash varphi\; \backslash lor\; \backslash psi)\; \backslash in\; \backslash Phi$ if and only if $\backslash varphi\; \backslash in\; \backslash Phi$ or $\backslash psi\; \backslash in\; \backslash Phi$, ## if $(\backslash varphi\backslash to\backslash psi)\; \backslash in\; \backslash Phi$ and $\backslash varphi\; \backslash in\; \backslash Phi$, then $\backslash psi\; \backslash in\; \backslash Phi$, ## $\backslash exists\; x\; \backslash ,\; \backslash varphi\; \backslash in\; \backslash Phi$ if and only if there is a term $t$ such that $\backslash varphi\backslash in\backslash Phi$.

Henkin's theorem

Let $S$ be a set of symbols. Let $\backslash Phi$ be a maximally consistent set of $S$-formulas containing witnesses. Define an equivalence relation $\backslash sim$ on the set of $S$-terms by $t\_0\; \backslash sim\; t\_1$ if $\backslash ;\; t\_0\; \backslash equiv\; t\_1\; \backslash in\; \backslash Phi$, where $\backslash equiv$ denotes equality. Let $\backslash overline\; t$ denote the equivalence class of terms containing $t$; and let $T\_\backslash Phi\; :=\; \backslash $ where $T^S$ is the set of terms based on the set of symbols $S$. Define the $S$-structure $\backslash mathfrak\; T\_\backslash Phi$ over $T\_\backslash Phi$, also called the term-structure corresponding to $\backslash Phi$, by: # for each $n$-ary relation symbol $R\; \backslash in\; S$, define $R^\; \backslash overline\; \backslash ldots\; \backslash overline$ if $\backslash ;\; R\; t\_0\; \backslash ldots\; t\_\; \backslash in\; \backslash Phi;$ # for each $n$-ary function symbol $f\; \backslash in\; S$, define $f^\; (\backslash overline\; \backslash ldots\; \backslash overline\; )\; :=\; \backslash overline\; ;$ # for each constant symbol $c\; \backslash in\; S$, define $c^:=\; \backslash overline\; c.$ Define a variable assignment $\backslash beta\_\backslash Phi$ by $\backslash beta\_\backslash Phi\; (x)\; :=\; \backslash bar\; x$ for each variable $x$. Let $\backslash mathfrak\; I\_\backslash Phi\; :=\; (\backslash mathfrak\; T\_\backslash Phi,\backslash beta\_\backslash Phi)$ be the term interpretation associated with $\backslash Phi$. Then for each $S$-formula $\backslash varphi$:

Sketch of proof

There are several things to verify. First, that $\backslash sim$ 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 $\backslash sim$ is an equivalence relation and also requires a proof that (1) and (2) are independent of the choice of $t\_0,\; \backslash ldots\; ,t\_$ class representatives. Finally, $\backslash mathfrak\; I\_\backslash Phi\; \backslash vDash\; \backslash varphi$ can be verified by induction on formulas.

See also

*Equiconsistency *Hilbert's problems *Hilbert's second problem *Jan Łukasiewicz *Paraconsistent logic *ω-consistency *Gentzen's consistency proof *Proof by contradiction

Footnotes

References

* 10th impression 1991. * * * (pbk.) * * *

External links

* {{Authority control Category:Proof theory Category:Hilbert's problems Category:Metalogic