This note is meant to collect a few basic facts about ordinals. I found that I didn’t really know how they work and this was embarrassing. I also couldn’t find any source that covered both the basics and some interesting applications, so I tried to assemble them here. Some results proved below are
- The Borel -algebra has the cardinality of the continuum
- (Sierpinski) There is a set in the plane such that every line intersects it in exactly two points
- (Cantor-Bendixson) Any closed subset of is a union of a perfect set and a countable set.
In what follows, many set-theoretical subtleties are ignored. In particular, von Neumann’s construction of ordinals is not presented.