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 $\preceq$:

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

Definition 2

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

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

• Equivalence of Definitions of Well-Ordering

• Definition:Well-Ordered Set
• Definition:Strict Well-Ordering

• Results about well-orderings can be found here.