# complete

### Algebra/Analysis

A linearly-ordered field is said to be complete if every subset with an upper bound has a least upper bound. Every complete ordered field is isomorphic to the set of real numbers.

A metric space $$X$$ is said to be complete if every Cauchy sequence in $$X$$ converges in $$X$$.

### Graph Theory

A simple graph is complete if every vertex is adjacent to (shares an edge with) every other vertex. The complete graph on $$n$$ vertices has $$\displaystyle\frac{n(n-1)}{2}$$ edges.

### Logic

A system of axioms for a mathematical theory is said to be complete if every theorem (i.e., true statement) of the theory is deducible from the axioms. The Gödel Incompleteness Theorem showed that any axiom system that includes arithmetic is necessarily incomplete.

### Set Theory

If $$F$$ is a filter on a set $$X$$ and $$\kappa$$ is a regular, uncountable cardinal, then we say that $$F$$ is $$\kappa$$-complete or $$\kappa$$-closed if $$\bigcap A \in F$$ for every $$A \subset F$$ with $$|A|<\kappa$$. Every filter is $$\omega$$-complete. If $$\kappa = \aleph_1$$ then $$F$$ if called countably complete.

A lattice is complete if every subset has a least upper bound and a greatest lower bound.