Completeness

Completeness

The term complete with respect to meta-logic is used to describe a system of logic for which every logical implication in that system is provable. In other words, if something true can be expressed in that system, it is provable in that system.

Formally, a logic is complete when for every Γ ⊨ φ, it is also true that Γ ⊢ φ.

A complete logical system leaves no logical truth that can be formulated within it unprovable, or undecidable. For instance, a system of number theory for the natural numbers, which contains the language elements needed to express 2 + 2 = 4, but cannot be used to express 2 + 2 = 4 is incomplete.

Completeness is an important property of a logical system, but unlike soundness, it is not necessary for a logical system to be complete to be useful. For example, second-order logic is sound but not complete. Kurt Gödel famously provided incompleteness theorems for systems of mathematical logic.

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