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Bibliography
1. General Expositions
Girard, J. Y.: 1981, ‘Π 12 -logic, part I: dilators’, Ann. Math. Log. 21, 75–219. (The basic text on dilators.)
Girard, J. Y.: ‘Proof-theory and logical complexity’, to appear: at editions Bibliopolis, Napoli. (Chapters 8–12 of this book present the most systematic available description of Π 12 -logic.)
Girard, J. Y. and H. R. Jervell: ‘Π 12 -logic’, in preparation for editions North Holland. (This book will follow the viewpoint of trees: all kinds of dendroids ...)
2. Relations to Generalized Recursion
Van de Wiele, J.: 1982, ‘Recursive dilators and generalized recursions’, in Stern (ed.) Proc. Herbrand Symp., North Holland. (This paper gives an equivalence between two notions of generalized recursion, by means of dilators; these notions were not thought as equivalent before.)
Girard, J. Y. and D. Normann: ‘Set-recursion and Π 12 -logic’, submitted for publication. (This paper gives an introduction to Π 12 -logic, viewed from the standpoint of recursion theory; it also gives a relativization of Van de Wiele's result, and as a corollary, we obtain section 8.2.)
Girard, J. Y. and J. Vauzeilles: ‘Les premiers récursivement inaccessible et Mahlo et la théorie des dilatateurs’. (Here we find a direct proof of section 8.2, and the proof of a similar result for the 1st rec. Mahlo.)
Girard, J. Y. and J.P. Ressayre: Eléments de logique Π 1 n . (Here the theory of ptykes of finite type is developed in the framework of undiscernability theory; relations between Π 1 n ordinals and the ptykes Ξ n are systematically studied.)
Ressayre, J. P.: 1982, ‘Bounding generalized recursive functions of ordinals by effective functors; a complement to the Girard theorem’, in Stern (ed.), Proc. Herbrand Symp. (This paper gives the most detailed account of the results which are around Theorem 8.1; for instance the fact that Theorem 8.1 holds cofinally in σo has been first proved here.)
3. Dilators and Related Concepts
Boquin, D.: ‘Regular dilators’, submitted. (This paper develops an alternative concept of dilators: the regularity condition on dilators renders them more effective ...)
Jervell, H. R.: 1982, ‘Introducing homogeneous trees’, in Stern (ed.), Proc. Herbrand Symp. (The concept of homogeneous tree turns out to be equivalent to the concept of ladder; homogeneous tress can be a nice way to start Π 12 -logic. But the concept is not as flexible as dilators.)
Masseron, M.: ‘Rungs and trees’, to appear in J.S.L. (A proof of the equivalence between ladders and homogeneous trees.)
Vauzeilles, J.: 1982, ‘Functors and ordinal notations III: dilators and gardens’, in Stern (ed.), Proc. Herbrand Symp. (Gardens are the initial concept with which Π 12 logic was developed; they were replaced by dilators for questions of simplicity, but they are basically richer. In this paper, an isomorphism between dilators and (simplified) gardens is given.)
4. Cut-elimination
Ferbus, M. C.: ‘Functorial bounds for cut-elimination in \(\mathbb{L}_{\beta \omega }\)’, submitted. (The calculus \(\mathbb{L}_{\beta \omega }\) is the Π 12 analogue of \(\mathbb{L}_{\omega 1\omega } \); here the familiar bounds for (usual) cut-elimination in \(\mathbb{L}_{\omega 1\omega } \) are made functorial.)
Girard, J. Y. and M. Masseron: ‘Proof-theoretic investigations of inductive definitions. Part II: monotonic definitions’, in preparation. (The cut-elimination theorem for inductive logic, in the case of positive operators.)
5. Λ and Related Topics
Girard, J. Y. and J. Vauzeilles: ‘Functors and ordinal notations; part I: a functorial construction of the Veblen hierarchy, part II: a functorial construction of the Bachmann hierarchy’, submitted. (These papers present an explicit relation between Λ and the more traditional Veblen and Bachmann methods.)
Schmerl, U. R.: 1982, ‘Number theory and the Bachmann-Howard ordinal’, in Stern (ed.), Proc. Herbrand Symp., North Holland. (Another very interesting relation between PA and the Howard ordinal, obtained by considering an extremely weak form of transfinite induction.)
Abrusci, M., J. Y. Girard and J. Van de Wiele: Some use of dilators in combinatorial problems’, in preparation. (This paper presents a proof of Goodstein's theorem by means of dilators, together with explicit bounds given in terms of the hierarchy λ; in a second part, it gives a solution to the “inverse Goodstein problem”, by means of a Bachmann-type hierarchy: here the number of steps is ηo, hence not provably terminating in ID 1!)
Päppinghaus, P: ‘Gödel's T and the Bachmann-Howard ordinal’, in preparation. (The objects of T can be considered as ptykes of the corresponding types; this enables one to associate ordinals to objects of type o → o in a natural way; in this paper, the heights of the associated ordinals are computed, and, as expected, there sup is shown to be the Howard ordinal.)
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Girard, JY. Introduction to 1Π2 -logic. Synthese 62, 191–216 (1985). https://doi.org/10.1007/BF00486046
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DOI: https://doi.org/10.1007/BF00486046