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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.

Information

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

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