# Definition:Well-Ordering/Class Theory

## Definition

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 known as

There is a school of thought that suggests that this may be referred to as a strong well-ordering, keeping the terminology different from that of a well-ordering.

The difference is that for an ordering to be a well-ordering, every (non-empty) subset must have a smallest element, whereas for an ordering to be a strong well-ordering, not only every subset but in fact every subclass must have a smallest element.

## 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.