Similarly, a class of serial relations, called well-ordered serial relations, can be defined, such that their corresponding relation-numbers include the ordinary finite **ordinals**, but also include relation-numbers which have many properties like those of the finite **ordinals**, though the fields of the relations belonging to them are not finite.

The definitions of the finite **ordinals** can be expressed without use of the corresponding cardinals, so there is no essential priority of cardinals to **ordinals**.

Here also it can be seen that the science of the finite **ordinals** is a particular subdivision of the general theory of classes and relations.

Contrasting the above definitions of number, cardinal and **ordinals**, with the alternative theory that number is an ultimate idea incapable of definition, we notice that our procedure exacts a greater attention, combined with a smaller credulity; for every idea, assumed as ultimate, demands a separate act of faith.

Are used sometimes as **ordinals**, i.e.

In the case of lists and schedules the numbers are only **ordinals**; but in the case of mathematical or statistical tables they are usually regarded as cardinals,though, when they represent values of a continuous quantity, they must be regarded as **ordinals** (§§ 26, 93).

Thus (writing **ordinals** in light type, and cardinals in heavy type) 9 comes after 4, and therefore 9 is greater than 4.

Owing to the correspondence between the finite cardinals and the finite **ordinals**, the propositions of cardinal arithmetic and ordinal arithmetic correspond point by point.