Abstract : Per Martin-Löf 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 paper 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 rôle 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 α ≤ β or β≤ α 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.