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Equalization of finite flowers

  • Stefano Berardi (a1)

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A dilator D is a functor from ON to itself commuting with direct limits and pull-backs. A dilator D is a flower iff D(x) is continuous. A flower F is regular iff F(x) is strictly increasing and F(f)(F(z)) = F(f(z)) (for f ϵ ON(x,y), z ϵ X).

Equalization is the following axiom: if F, G ϵ Flr (class of regular flowers), then there is an H ϵ Flr such that F ° H = G ° H. From this we can deduce that if is a set ⊆ Flr, then there is an H ϵ Flr which is the smallest equalizer of (it can be said that H equalizes ℱ iff for every F, G ϵ we have F ° H = G ° H). Equalization is not provable in set theory because equalization for denumerable flowers is equivalent to -determinacy (see a forthcoming paper by Girard and Kechris).

Therefore it is interesting to effectively find, by elementary means, equalizers even in the simplest cases. The aim of this paper is to prove Girard and Kechris's conjecture: “ is the (smallest) equalizer for Flr < ω” (where Flr < ω denotes the set of finite regular flowers). We will verify that is an equalizer of Flr < ω; we will sketch the proof that it is the smallest one at the end of the paper. We will denote by H.

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References

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Girard, J.-Y., -logic. Part 1: Dilators, Annals of Mathematical Logic, vol. 21 (1981), pp. 75219.
Girard, J.-Y., A survey of -logic, Logic, methodology and philosophy of science VI (Hannover, 1979), North-Holland, Amsterdam, 1982, pp. 89107.
Girard, J.-Y., Note manuscrite sur l’axiome d’égalisation, 1985 (unpublished).
Girard, J.-Y., Note manuscrite sur: l’éegalisation implique la détermination , 1985 (unpublished).
Girard, J.-Y., Proof theory and logical complexity, Chapters 8–12, Bibliopolis, Naples (to appear).

For a more careful study on -logic, see:

Girard, J.-Y. and Normann, D., Set recursion and -logic, Annals of Pure and Applied Logic, vol. 28 (1985), pp. 255286.
Girard, J.-Y. and Vauzeilles, J., Functors and ordinal notations. I: A functorial construction of the Veblen hierarchy, this Journal, vol. 49 (1984), pp. 713729.
Girard, J.-Y. and Vauzeilles, J., Functors and ordinal notations. II: A functorial construction of the Bachmann hierarchy, this Journal, vol. 49 (1984), pp. 10791114.
Girard, J.-Y. and Vauzeilles, J., Les premiers récursivement inaccessibles et Mahlo et la théorie des dilatateurs, Archiv für Marthematische Logik und Grundlagenforschung, vol. 24 (1981), pp. 167191.
Masseron, M., Majoration des fonctions -récursives par des (ω)-échelles primitives récursives, Thèse du troisième cycle, Université Paris-Nord, Paris, 1980.
Masseron, M., Rungs and trees, this Journal, vol. 48 (1983), pp. 847863.
Van de Wiele, J., Recursive dilators and generalized recursions, Proceedings of the Herbrand symposium (Marseilles, 1981), North-Holland, Amsterdam, 1982, pp. 325332.
Vauzeilles, J., Functors and ordinal notations. III: Dilators and gardens, Proceedings of the Herbrand symposium (Marseilles, 1981), North-Holland, Amsterdam, 1982, pp. 333364.
Girard, J.-Y. and Ressayre, J.-P.,Éléments de logique , Recursion theory, Proceedings of Symposia in Pure Mathematics, vol. 42, American Mathematical Society, Providence, Rhode Island, 1985, pp. 389445.

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