L'objet de ce texte est de montrer que des fonctions qui appartiennent à une famille vérifiant une propriété de finitude a priori non uniforme sont en fait définissables dans une structure o-minimale.

I show that any sentence of nth-order (pure or applied) arithmetic can be expressed with no loss of compositionality as a second-order sentence containing no arithmetical vocabulary, and use this result to prove a completeness theorem for applied arithmetic. More specifically. I set forth an enriched second-order language L. a sentence of L (which is true on the intended interpretation of L), and a compositionally recursive transformation Tr defined on formulas of L, and show that they have the following two properties: (a) in a universe with at least ℶn−2 objects, any formula of nth-order (pure or applied) arithmetic can be expressed as a formula of L, and (b) for any sentence of is a second-order sentence containing no arithmetical vocabulary, and