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In [[New Foundations]] (NFU), the '''order type''' of a well-ordering ''W'' is the set of all well-orderings which are similar to ''W''. The set of '''ordinal numbers''' is the set of all order types of well-orderings.
This does not work in [[ZFC]], because the equivalence classes are too large. It would be formally possible to use [[Scott's trick]] to define the ordinals in essentially the same way, but a device of [[John von Neumann|von Neumann]] is more commonly used.
For any partial order <math>\leq</math>, the corresponding '''strict partial order''' < is defined as <math>\{(x,y) \mid x \leq y \wedge x \neq y\}</math>. Strict linear orders and strict well-orderings are defined similarly.
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