# Definition:Well-Ordering

## Definition

Let $\struct {S, \preceq}$ 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 $\preceq$:

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

### Definition 2

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

### Class Theory

In the context of class theory, the definition follows the same lines:

Let $V$ be a basic universe.

Let $\RR \subseteq V \times V$ be a total ordering.

Then $\RR$ is a well-ordering if and only if:

every non-empty subclass of $\Field \RR$ has a smallest element under $\RR$

where $\Field \RR$ denotes the field of $\RR$.

## Also defined as

1955: John L. Kelley: General Topology uses the term well-ordering to mean what $\mathsf{Pr} \infty \mathsf{fWiki}$ calls a strict strong well-ordering.

1980: Kenneth Kunen: Set Theory: An Introduction to Independence Proofs uses the term well-ordering to mean what $\mathsf{Pr} \infty \mathsf{fWiki}$ calls a strict well-ordering.

## Also see

• Results about well-orderings can be found here.