No CrossRef data available.
Published online by Cambridge University Press: 12 March 2014
In a recent work on -recursivity ([M] in which the principal result of [G.2] is obtained by more direct methods) I have given a very simple lemma on the structure of the so-called rungs [G.1] which enables us to describe precisely the relation they have with Jervell's tree [J] in the wellfounded case (R.O. Gandy has also worked out this relation in the wellfounded case, but with techniques very different from those introduced here).
In the present work I treat the general case in §§I and II. The results are summarized in §III. These results are then exploited by means of the operator Σ*, which is the principal tool of Lemma (4.1) (i.e. the generalized version of the previous structural lemma). A systematical use of Σ* is also the basis of the study of ladders in the last section.
Recall that the notions of rungs and ladders were introduced by J.-Y. Girard [G, 0] and [G.1] as a basic tool for ordinal analysis of Π½-logic (i.e. roughly speaking, logic with β-rule) and the introduction of trees and homogeneous trees by H.R. Jervell is a first attempt to give to these notions a geometrical interpretation without any explicit mention of functors and categorical limits. In fact, as it is evident in the last section of the present paper, limits play a modest role in the study of ladders, but their properties with respect to limits are of fundamental interest in their applications (a domain which is out of the scope of this basic work. Readers interested in applications to ordinal analysis are invited to consult the references). Note that my trees are not exactly those introduced by H.R. Jervell: his extra symbols λ and ρ are respectively changed into 0 and the type of the concerned tree. More important is the fact that we consider only the extremal nodes of the trees. The objects so obtained are then as close as possible to Girard's dendroids.
I am indebted to J.-Y. Girard for his advice about my work.
To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about saving to your Kindle.
Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.
Find out more about the Kindle Personal Document Service.
To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.
To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.