Definition:Well-Ordering

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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.