Full Idea
The collection of ordinals is defined inductively: Basis: the empty set is an ordinal; Ind: for an ordinal x, the union with its singleton is also an ordinal; and any arbitrary (possibly infinite) union of ordinals is an ordinal.
Gist of Idea
Ordinals are the empty set, union with the singleton, and any arbitrary union of ordinals
Source
Michal Walicki (Introduction to Mathematical Logic [2012], 2.3)
Book Reference
Walicki,Michal: 'Introduction to Mathematical Logic' [World Scientific 2012], p.88
A Reaction
[symbolism translated into English] Walicki says they are called 'ordinal numbers', but are in fact a set.
Related Idea
Idea 17756 The union of finite ordinals is the first 'limit ordinal'; 2ω is the second... [Walicki]