Ordinals numbers, or transfinite numbers, are a set-theoretic notion allowing to extend the ordered structure of natural numbers beyond infinity. In fact, the collection of ordinals is bigger than any set, and thus contains more elements than we could ever describe with finite sentences.
Still, one can wonder how many ordinals can be expressed with finite sentences. Ordinal notations are systems of syntax that, similar to how one could describe mathematical objects with words, describe with formal terms a given set of ordinals.
In this talk, I will present ordinal notations, an example of their use and describe the notation of Buchholz's psi functions, allowing to define a lot of important countable ordinals.