Difference between revisions of "Consistent"
m (Text replacement - "http://" to "https://") |
|||
| Line 1: | Line 1: | ||
[[File:lighterstill.jpg]][[File:Consistency.jpg|center|frame]] | [[File:lighterstill.jpg]][[File:Consistency.jpg|center|frame]] | ||
| − | *[ | + | *[https://en.wikipedia.org/wiki/16th_century 1594] |
==Definitions== | ==Definitions== | ||
*1a archaic : condition of [[adhering]] [[together]] : firmness of [[material]] substance | *1a archaic : condition of [[adhering]] [[together]] : firmness of [[material]] substance | ||
| Line 10: | Line 10: | ||
==Description== | ==Description== | ||
| − | In [[logic]], a '''consistent''' [[theory]] is one that does not contain 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]]; this is the sense used in traditional [ | + | In [[logic]], a '''consistent''' [[theory]] is one that does not contain 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]]; this is the sense used in traditional [https://en.wikipedia.org/wiki/Term_logic Aristotelian logic], although in contemporary [[mathematical]] [[logic]] the term satisfiable is used instead. The [[syntactic]] definition states that a [[theory]] is consistent if there is no [[formula]] P such that both P and its negation are provable from the [[axioms]] of the theory under its associated deductive system. |
| − | If these [[semantic]] and [[syntactic]] definitions are [[equivalent]] for a particular [[logic]], the logic is complete. The completeness of [ | + | If these [[semantic]] and [[syntactic]] definitions are [[equivalent]] for a particular [[logic]], the logic is complete. The completeness of [https://en.wikipedia.org/wiki/Sentential_calculus sentential calculus] was proved by [https://en.wikipedia.org/wiki/Paul_Bernays Paul Bernays] in 1918 and [https://en.wikipedia.org/wiki/Emil_Post Emil Post] in 1921, while the completeness of [https://en.wikipedia.org/wiki/Predicate_calculus predicate calculus] was proved by [https://en.wikipedia.org/wiki/Kurt_G%C3%B6del Kurt Gödel] in 1930, and consistency [[proofs]] for arithmetics restricted with respect to the [https://en.wikipedia.org/wiki/Induction induction axiom schema] were proved by Ackermann (1924), von Neumann (1927) and Herbrand (1931. Stronger logics, such as [https://en.wikipedia.org/wiki/Second-order_logic second-order logic], are not complete. |
| − | A consistency [[proof]] is a [[mathematical]] proof that a particular [[theory]] is consistent. The early [[development]] of mathematical [ | + | A consistency [[proof]] is a [[mathematical]] proof that a particular [[theory]] is consistent. The early [[development]] of mathematical [https://en.wikipedia.org/wiki/Proof_theory proof theory] was driven by the [[desire]] to provide finitary consistency proofs for all of mathematics as part of [https://en.wikipedia.org/wiki/Hilbert%27s_program Hilbert's program]. Hilbert's program was strongly impacted by [https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems 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 [ | + | 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 [https://en.wikipedia.org/wiki/Cut-elimination cut-elimination] (or equivalently the [https://en.wikipedia.org/wiki/Normalization_property normalization] of the [https://en.wikipedia.org/wiki/Curry-Howard 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== | ==Consistency and completeness in arithmetic== | ||
| − | In theories of arithmetic, such as [ | + | In theories of arithmetic, such as [https://en.wikipedia.org/wiki/Peano_arithmetic 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. |
| − | [ | + | [https://en.wikipedia.org/wiki/Presburger_arithmetic Presburger arithmetic] is an [[axiom]] [[system]] for the natural numbers under addition. It is both consistent and complete. |
| − | [ | + | [https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems Gödel's incompleteness theorems] show that any sufficiently strong effective theory of arithmetic cannot be both complete and consistent. Gödel's theorem applies to the theories of Peano arithmetic (PA) and [https://en.wikipedia.org/wiki/Primitive_recursive_arithmetic Primitive recursive arithmetic] (PRA), but not to Presburger arithmetic. |
| − | Moreover, Gödel's second incompleteness theorem shows that the consistency of sufficiently strong effective 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, effective, consistent theory of arithmetic can never be proven in that system itself. The same result is true for effective theories that can describe a strong enough fragment of arithmetic – including set theories such as [ | + | Moreover, Gödel's second incompleteness theorem shows that the consistency of sufficiently strong effective 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, effective, consistent theory of arithmetic can never be proven in that system itself. The same result is true for effective theories that can describe a strong enough fragment of arithmetic – including set theories such as [https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory Zermelo–Fraenkel set theory]. These set theories cannot prove their own Gödel sentences – provided that they are consistent, which is generally believed. |
[[Category: Logic]] | [[Category: Logic]] | ||
Latest revision as of 23:45, 12 December 2020
Definitions
- b : firmness of constitution or character : persistency
- 2: degree of firmness, density, viscosity, or resistance to movement or separation of constituent particles <boil the juice to the consistency of a thick syrup>
- 3a : agreement or harmony of parts or features to one another or a whole : correspondence; specifically : ability to be asserted together without contradiction
- b : harmony of conduct or practice with profession <followed her own advice with consistency>
Description
In logic, a consistent theory is one that does not contain 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; this is the sense used in traditional Aristotelian logic, although in contemporary mathematical logic the term satisfiable is used instead. The syntactic definition states that a theory is consistent if there is no formula P such that both P and its negation are provable from the axioms of the theory under its associated deductive system.
If these semantic and syntactic definitions are equivalent for a particular logic, the logic is complete. The completeness of 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 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
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 effective 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 effective 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, effective, consistent theory of arithmetic can never be proven in that system itself. The same result is true for effective 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 sentences – provided that they are consistent, which is generally believed.
