Constructive theory of ordinals

Book chapter, 2023

Martin-Löf [1] describes recursively constructed ordinals. He gives a constructively acceptable version of Kleene's computable ordinals. In fact, the Turing definition of computable functions is not needed from a constructive point of view.We give in this chapter a constructive theory of ordinals that is similar to Martin-Löf's theory, but based only on the two relations "x ≤ y" and "x < y", i.e., without considering sequents whose intuitive meaning is a classical disjunction. In our setting, the operation "supremum of ordinals" plays an important role through its interactions with the relations "x ≤ y" and "x < y". This allows us to approach as much as we may the notion of linear order when the property "A ≤ B or B ≤ A" is provable only within classical logic. Our aim is to give a formal definition corresponding to intuition, and to prove that our constructive ordinals satisfy constructively all desirable properties.