Where does ordinal collide?

In statistics an ordinal scale is a level of measurement whose values carry a meaningful order but not meaningful distances, so categories can be ranked yet the gaps between them cannot be treated as equal or added. Stevens's 1946 On the theory of scales of measurement defined the ordinal scale within his nominal-ordinal-interval-ratio hierarchy and specified which statistics it permits. The referent is a property of measured data, how much arithmetic its ordering supports, and it is distinct from an infinite order type in set theory or a word like third, sharing only the idea of rank.[1]

Unit

A measurement scale permitting rank order but not distance

Recovery means

Not applicable

Measured by

Identified by whether categories can be ranked while differences between them lack a fixed meaning

In set theory an ordinal number is an order type of a well-ordered set, a generalisation of the counting numbers that continues transfinitely past the finite ones into the infinite ordinals such as omega and beyond, each ordinal being the set of all smaller ordinals. Halmos's Naive Set Theory constructs the ordinals and their arithmetic of order. The referent is a rigorously defined object capturing position in a well-ordering, an abstract number concept unrelated to a data scale or a grammatical numeral apart from the shared root meaning of order.[2]

Unit

An order type of a well-ordered set (finite or transfinite)

Recovery means

Not applicable

Measured by

Defined and compared by order-isomorphism of the well-ordered sets they represent