Ordinals

We will define ordinals . The main point is actually that we can do without them.

Definition

A set is transitive, , if and only if each element is also a subset. In other words, .

Definition

A set is an ordinal, , if and only if and for all , .

Exercise

Define . Show that if then . Note that , the null set, is an ordinal.

We know what happens if we talk about the set of all sets - Russell's paradox is waiting for you. Ordinals give us a neat way to get around the problem of big sets - we call all sets a class. After that - all classes are called a family. This hierarchy may go on forever. We will not use ordinals further, except a little in the next paragraph, where we talk a bit about set-based mappings.