Consistant vs. Consistent

Consistent

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

T

{\displaystyle T}

is consistent if and only if there is no formula

φ

{\displaystyle \varphi }

such that both

φ

{\displaystyle \varphi }

and its negation

¬

φ

{\displaystyle \lnot \varphi }

are elements of the set

T

{\displaystyle T}

. Let

A

{\displaystyle A}

be a set of closed sentences (informally "axioms") and

⟨

A

⟩

{\displaystyle \langle A\rangle }

the set of closed sentences provable from

A

{\displaystyle A}

under some (specified, possibly implicitly) formal deductive system. The set of axioms

A

{\displaystyle A}

is consistent when

⟨

A

⟩

{\displaystyle \langle A\rangle }

is.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.

Consistent (adjective)

Possessing firmness or fixedness; firm; hard; solid.

Consistent (adjective)

Having agreement with itself or with something else; having harmony among its parts; possesing unity; accordant; harmonious; congruous; compatible; uniform; not contradictory.

Consistent (adjective)

Living or acting in conformity with one's belief or professions.

Consistent (adjective)

(sometimes followed by `with') in agreement or consistent or reliable;

"testimony consistent with the known facts"

"I have decided that the course of conduct which I am following is consistent with my sense of responsibility as president in time of war"

Consistent (adjective)

marked by an orderly, logical, and aesthetically consistent relation of parts;

"a logical argument"

"the orderly presentation"

Consistent (adjective)

capable of being reproduced;

"astonishingly reproducible results can be obtained"

Consistent (adjective)

the same throughout in structure or composition;

"bituminous coal is often treated as a consistent and homogeneous product"