Teljességétl

Teljességétl, also known as completeness, is a fundamental concept in logic, mathematics, and computer science, referring to the state of being whole or entire, without any missing parts. In logic, a set of axioms or rules is considered complete if it can prove or disprove every statement within its domain. For example, the axioms of Euclidean geometry are complete for proving or disproving geometric statements within that system.

In mathematics, a set is complete if it contains all its limit points. For instance, the set of rational numbers is not complete because it lacks the irrational numbers, which are necessary for the convergence of certain sequences. The set of real numbers, however, is complete because it includes all its limit points.

In computer science, completeness is often discussed in the context of algorithms and programming languages. An algorithm is complete if it can solve all instances of a given problem. A programming language is complete if it can express all computable functions.

The concept of completeness is closely related to that of consistency. A set of axioms or rules is consistent if it does not contain any contradictions. However, a set can be consistent and yet incomplete, meaning it can prove or disprove some statements but not all. Conversely, a set can be complete and yet inconsistent, meaning it contains contradictions and thus cannot be used to prove or disprove any statements.