In classical deductive logic, a consistent theory is one that does not lead to a logical contradiction.
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 – ]" cf .
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 .
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.
(Please note 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.
The completeness of the sentential calculus was proved by Paul Bernays in 1918 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 .
Also .
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 .
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 .
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 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{Con} \Phi) if there is no formula \varphi such that \Phi \vdash \varphi and \Phi \vdash \lnot\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 for every formula \varphi, if \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
The following are equivalent:
\operatorname{Inc}\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{I} such that \mathfrak{I} \vDash \Phi .
For all \Phi and \varphi:
if not  \Phi \vdash \varphi, then \operatorname{Con}\left( \Phi \cup \{\lnot\varphi\}\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).
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{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.
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:
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.
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}};
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:
\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_{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
\{\varphi,\varphi'\}\not\subseteq Taccording to De Morgan's laws
\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
Footnotes
References
10th impression 1991.
(pbk.)
External links
