# 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**.

## Sources

- 2010: Raymond M. Smullyan and Melvin Fitting:
*Set Theory and the Continuum Problem*(revised ed.) ... (previous) ... (next): Chapter $4$: Superinduction, Well Ordering and Choice: Part $\text I$ -- Superinduction and Well Ordering: $\S 1$ Introduction to well ordering