Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-25T19:36:02.681Z Has data issue: false hasContentIssue false
  • English
  • Français

Equalization of finite flowers

Published online by Cambridge University Press:  12 March 2014

Stefano Berardi
Stefano Berardi*
Affiliation:
Via Susa 23, 10138 Torino, Italy
Get access Rights & Permissions [Opens in a new window]

Extract

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 ℱ is a set ⊆ Flr, then there is an H ϵ Flr which is the smallest equalizer of ℱ (it can be said that H equalizes ℱ iff for every F, G ϵ ℱ 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.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 53 , Issue 1 , March 1988 , pp. 105 - 123
Copyright
Copyright © Association for Symbolic Logic 1988

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Girard, J.-Y., -logic. Part 1: Dilators, Annals of Mathematical Logic, vol. 21 (1981), pp. 75–219.CrossRefGoogle Scholar
Girard, J.-Y., A survey of -logic, Logic, methodology and philosophy of science VI (Hannover, 1979), North-Holland, Amsterdam, 1982, pp. 89–107.Google Scholar
Girard, J.-Y., Note manuscrite sur l’axiome d’égalisation, 1985 (unpublished).Google Scholar
Girard, J.-Y., Note manuscrite sur: l’éegalisation implique la détermination , 1985 (unpublished).Google Scholar
Girard, J.-Y., Proof theory and logical complexity, Chapters 8–12, Bibliopolis, Naples (to appear).Google Scholar
Girard, J.-Y. and Normann, D., Set recursion and -logic, Annals of Pure and Applied Logic, vol. 28 (1985), pp. 255–286.CrossRefGoogle Scholar
Girard, J.-Y. and Vauzeilles, J., Functors and ordinal notations. I: A functorial construction of the Veblen hierarchy, this Journal, vol. 49 (1984), pp. 713–729.Google Scholar
Girard, J.-Y. and Vauzeilles, J., Functors and ordinal notations. II: A functorial construction of the Bachmann hierarchy, this Journal, vol. 49 (1984), pp. 1079–1114.Google Scholar
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. 167–191.CrossRefGoogle Scholar
Masseron, M., Majoration des fonctions -récursives par des (ω)-échelles primitives récursives, Thèse du troisième cycle, Université Paris-Nord, Paris, 1980.Google Scholar
Masseron, M., Rungs and trees, this Journal, vol. 48 (1983), pp. 847–863.Google Scholar
Van de Wiele, J., Recursive dilators and generalized recursions, Proceedings of the Herbrand symposium (Marseilles, 1981), North-Holland, Amsterdam, 1982, pp. 325–332.CrossRefGoogle Scholar
Vauzeilles, J., Functors and ordinal notations. III: Dilators and gardens, Proceedings of the Herbrand symposium (Marseilles, 1981), North-Holland, Amsterdam, 1982, pp. 333–364.CrossRefGoogle Scholar
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. 389–445.CrossRefGoogle Scholar