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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 \varphi such that both \varphi and its negation \lnot\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 \varphi, \lnot \varphi \in \langle A \rangle for no formula \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

\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 \operatorname \Phi) if there is no formula \varphi such that \Phi \vdash \varphi and \Phi \vdash \lnot\varphi. Otherwise \Phi is inconsistent (written \operatorname\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 for every formula \varphi, if \operatorname (\Phi \cup \) 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 ) \in \Phi, where \varphi denotes the substitution of each x in \varphi by a t; see also First-order logic.

Basic results

# The following are equivalent: ## \operatorname\Phi ## For all \varphi,\; \Phi \vdash \varphi. # Every satisfiable set of formulas is consistent, where a set of formulas \Phi is satisfiable if and only if there exists a model \mathfrak such that \mathfrak \vDash \Phi . # For all \Phi and \varphi: ## if not \Phi \vdash \varphi, then \operatorname\left( \Phi \cup \\right); ## if \operatorname\Phi and \Phi \vdash \varphi, then \operatorname \left(\Phi \cup \\right); ## if \operatorname\Phi, then \operatorname\left( \Phi \cup \\right) or \operatorname\left( \Phi \cup \\right). # 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 (\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\in\Phi.

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 := \ 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: # for each n-ary relation symbol R \in S, define R^ \overline \ldots \overline if \; R t_0 \ldots t_ \in \Phi; # for each n-ary function symbol f \in S, define f^ (\overline \ldots \overline ) := \overline ; # for each constant symbol c \in S, define c^:= \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:
\mathfrak I_\Phi \vDash \varphi if and only if \; \varphi \in \Phi.


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_ 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, 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 #\\not\subseteq Taccording to De Morgan's laws #\varphi'\not\in T \lor \varphi\not\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