# 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.