# A model for barrecursion of higher types

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A model for barrecursion of higher types COMPOSITIO MATHEMATICA B. SCARPELLINI Amodel for barrecursion of higher types Compositio Mathematica, tome 23, no 1 (1971), p. 123-153. <http://www.numdam.org/item?id=CM_1971__23_1_123_0> © Foundation Compositio Mathematica, 1971, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http:// http://www.compositio.nl/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commer- ciale ou impression systématique est constitutive d’une infraction pé- nale. Toute copie ou impression de ce fichier doit contenir la pré- sente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ 123 A MODEL FOR BARRECURSION OF HIGHER TYPES by B. Scarpellini COMPOSITIO MATHEMATICA, Vol. 23, Fasc. 1, 1971, pag. 123-153 Wolters-Noordhoff Publishing Printed in the Netherlands Introduction Problems connected with the foundations of mathematics led C. Spec- tor to consider a certain kind of functional equation. The solution of this functional equation is provided by a certain principle, the ’principle of barrecursion". One problem with respect to this functional equation has up to now remained open, namely to find a family F of functionals with the property: if the parameters of the equation belong to F then there is a solution which belongs to F. There is so to speak a weak and a strong ver- sion of this problem: a) the weak version is that one given above, b) the strong version requires that the elements of F are constructive in one sense or the other. Here we propose a solution of the weak problem. More precisely, we construct two families S’ and K. The first is in essence already described in [2] but it has the disadvantage that its construction leads beyond classical analysis. The second, K, is a more elaborate version of S and its construction remains within the scope of

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