# Definition:Well-Ordering

## Definition

Let $\left({S, \preceq}\right)$ be an ordered set.

### Definition 1

The ordering $\preceq$ is a well-ordering on $S$ if and only if every non-empty subset of $S$ has a smallest element under $\preceq$:

$\forall T \subseteq S: \exists a \in T: \forall x \in T: a \preceq x$

### Definition 2

The ordering $\preceq$ is a well-ordering on $S$ if and only if $\preceq$ is a well-founded total ordering.

## Also defined as

1955: John L. Kelley: General Topology uses this term to mean what we call a strict strong well-ordering.

1980: Kenneth Kunen: Set Theory: An Introduction to Independence Proofs uses this term for what we call a strict well-ordering.

## Also see

• Results about well-orderings can be found here.