In classical deductive 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 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
Contents
- Consistency and completeness in arithmetic and set theory
- Notation
- Definition
- Basic results
- Henkins theorem
- Sketch of proof
- Model theory
- References
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 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.
For all
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.
Notation
⊢ (Turnstile symbol) in the following context of Mathematical logic, means "provable from". That is, a ⊢ 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
Basic results
- The following are equivalent:
- Inc
Φ - For all
ϕ , Φ ⊢ ϕ .
- Inc
- Every satisfiable set of formulas is consistent, where a set of formulas
Φ is satisfiable if and only if there exists a modelI such thatI ⊨ Φ . - For all
Φ andϕ :- if not
Φ ⊢ ϕ , then Con( Φ ∪ { ¬ ϕ } ) ; - if Con
Φ andΦ ⊢ ϕ , then Con( Φ ∪ { ϕ } ) ; - if Con
Φ , then Con( Φ ∪ { ϕ } ) or Con( Φ ∪ { ¬ ϕ } ) .
- if not
- Let
Φ be a maximally consistent set of formulas and contain witnesses. For allϕ andψ :- if
Φ ⊢ ϕ , thenϕ ∈ Φ , - either
ϕ ∈ Φ or¬ ϕ ∈ Φ , -
( ϕ ∨ ψ ) ∈ Φ if and only ifϕ ∈ Φ orψ ∈ Φ , - if
( ϕ → ψ ) ∈ Φ andϕ ∈ Φ , thenψ ∈ Φ , -
∃ x ϕ ∈ Φ if and only if there is a termt such thatϕ t x ∈ Φ .
- if
Henkin's theorem
Let
Define a binary relation
Define the
- for
n -aryR ∈ S ,R T Φ t 0 ¯ … t n − 1 ¯ R t 0 … t n − 1 ∈ Φ ; - for
n -aryf ∈ S ,f T Φ ( t 0 ¯ … t n − 1 ¯ ) := f t 0 … t n − 1 ¯ - for
c ∈ S ,c T Φ := c ¯
Let
For all
Sketch of proof
There are several things to verify. First, that
Model theory
In ZFC set theory with classical first-order logic, an inconsistent theory
-
{ ϕ , ϕ ′ } ⊈ T -
ϕ ′ ∉ T ∨ ϕ ∉ T