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Embeddability of ptykes

Published online by Cambridge University Press:  12 March 2014

Jean-Yves Girard  and
Dag Normann
Jean-Yves Girard
Affiliation:
UFR de Mathématique, Université Paris-VII, 75251 Paris, France
Dag Normann
Affiliation:
Institute of Mathematics, University of Oslo, 0316, Oslo, Norway, E-mail: dnorman@ma-mail.uio.no
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Extract

The notion of a ptyx (plural, ptykes) is obtained by generalising the ordinals On and the dilators Dil introduced by Girard [1] into a typed hierarchy. The notion is due to Girard, and a detailed treatment will appear in Girard [2].

In this Introduction we will give a summary of the theory of ptykes. It will be an advantage to be familiar with the theory of dilators as introduced in Girard [1] or as presented in Girard and Normann [3].

The class On of ordinals is organised into a category ON by using strictly increasing maps f: x → y as morphisms. A dilator will be a functor F: ON → ON commuting with pullbacks and direct limits. Associated with each dilator F there is a denotation system DF, obtained as follows: For each x ∈ On and y ∈ F(x), y can be given a unique denotation (c; x0,…, xn−1;x)F, where c∈F(n) (n = {0,…, n − 1}), y = F(ϕ)(c) (where ϕ is defined by ϕ(i) = xi for i < n), and for no m < n and ψ: m → n do we have c ∈ im(ψ).

We let the trace Tr(F) be the set of pairs (c, n) occurring in a denotation.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 57 , Issue 2 , June 1992 , pp. 659 - 676
Copyright
Copyright © Association for Symbolic Logic 1992

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References

REFERENCES

[1]Girard, J.-Y., -logic. Part 1: Dilators, Annals of Mathematical Logic, vol. 21 (1981), pp. 75–219.CrossRefGoogle Scholar
[2]Girard, J.-Y., Proof theory and logical complexity, Vol. II, North-Holland, Amsterdam (to appear).Google Scholar
[3]Girard, J.-Y. and Normann, D., Set recursion and -logic, Annals of Pure and Applied Logic, vol. 28 (1985), pp. 255–286.CrossRefGoogle Scholar
[4]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
[5]Kechris, A. S., Boundedness theorems for dilators and ptykes, Annals of Pure and Applied Logic, vol. 52(1991), pp. 79–92.CrossRefGoogle Scholar
[6]Kechris, A. S. and Woodin, H., A strong boundedness of dilators, Annals of Pure and Applied Logic, vol. 52 (1991), pp. 93–98.CrossRefGoogle Scholar
[7]Normann, D., The recursion theory of ptykes, Oslo University Preprint Series, no. 7, Oslo, 1984.Google Scholar
[8]Päppinghaus, P., Rekursion über Dilatoren und die Bachmann-Hierarchie, Archive for Mathematical Logic, vol. 28 (1989), pp. 57–73.CrossRefGoogle Scholar
[9]Päppinghaus, P., Ptykes in Gödels T und Definierbarkeit von Ordinalzahlen, Archive for Mathematical Logic, vol. 28 (1989), pp. 119–141.CrossRefGoogle Scholar
[10]Rinvold, R., Denotation systems which generalize to ordinal numbers, Candidate's dissertation, University of Oslo, Oslo, 1988, (Norwegian)Google Scholar