Skip to main content Accessibility help
×
Home
Hostname: page-component-684899dbb8-gbqfq Total loading time: 0.242 Render date: 2022-05-27T07:36:13.862Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "useNewApi": true }

Accessible Recursive Functions

Published online by Cambridge University Press:  15 January 2014

Stanley S. Wainer*
Affiliation:
School of Mathematics, University of Leeds, Leeds LS2 9JT, UkE-mail:s.s.wainer@leeds.ac.uk

Abstract

The class of all recursive functions fails to possess a natural hierarchical structure, generated predicatively from “within”. On the other hand, many (proof-theoretically significant) sub-recursive classes do. This paper attempts to measure the limit of predicative generation in this context, by classifying and characterizing those (predictably terminating) recursive functions which can be successively defined according to an autonomy condition of the form: allow recursions only over well-orderings which have already been “coded” at previous levels. The question is: how can a recursion code a well-ordering? The answer lies in Girard's theory of dilators, but is reworked here in a quite different and simplified framework specific to our purpose. The “accessible” recursive functions thus generated turn out to be those provably recursive in ( –CA)0.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1999

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

[1] Arai, T., A slow growing analogue of Buchholz' proof, Annals of Pure and Applied Logic, vol. 54 (1991), pp. 101120.CrossRefGoogle Scholar
[2] Buchholz, W., An independence result for ( – CA) + (BI), Annals of Pure and Applied Logic, vol. 32 (1987), pp. 131155.CrossRefGoogle Scholar
[3] Buchholz, W., Cichon, E. A., and Weiermann, A., A uniform approach to fundamental sequences and subrecursive hierarchies, Mathematical Logic Quarterly, vol. 40 (1994), pp. 273286.CrossRefGoogle Scholar
[4] Buchholz, W., Feferman, S., Pohlers, W., and Sieg, W., Iterated inductive definitions and subsystems of analysis, Lecture Notes in Mathematics, no. 897, Springer-Verlag, Berlin, 1981.Google Scholar
[5] Fairtlough, M. and Wainer, S. S., Hierarchies of provably recursive functions, Handbook of proof-theory (Buss, S., editor), Studies in Logic, no. 137, North-Holland, Amsterdam, 1998, pp. 149207.CrossRefGoogle Scholar
[6] Feferman, S., Three conceptual problems that bug me, to appear in the Proceedings of the 7th Scandinavian Logic Symposium, Uppsala 1996.Google Scholar
[7] Girard, J-Y., -logic, Part I: Dilators, Annals of Mathematical Logic, vol. 21 (1981), pp. 75219.CrossRefGoogle Scholar
[8] Kadota, N., On Wainer's notation for a minimal subrecursive inaccessible ordinal, Mathematical Logic Quarterly, vol. 39 (1993), pp. 217227.CrossRefGoogle Scholar
[9] Rathjen, M., How to develop proof-theoretic ordinal functions on the basis of admissible ordinals, Mathematical Logic Quarterly, vol. 39 (1993), pp. 4754.CrossRefGoogle Scholar
[10] Schmidt, D., Built-up systems of fundamental sequences and hierarchies of number-theoretic functions, Arkiv für Math. Logic und Grundlagenforschung, vol. 18 (1976), pp. 4753.CrossRefGoogle Scholar
[11] Vauzeilles, J., Functors and ordinal notations IV: The Howard ordinal and the functorΛ, this Journal, vol. 50 (1985), pp. 331338.Google Scholar
[12] Wainer, S. S., Slow growing vs. fast growing, this Journal, vol. 54 (1989), pp. 608–614.Google Scholar
[13] Wainer, S. S., Accessible segments of the fast growing hierarchy, Proceedings of logic colloquium 95, Haifa (Makowsky, J. and Ravve, E., editors), Lecture Notes in Logic, no. 11, Springer-Verlag, Berlin, 1998, pp. 339348.Google Scholar
2
Cited by

Save article to Kindle

To save this article to your Kindle, first ensure coreplatform@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 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.

Accessible Recursive Functions
Available formats
×

Save article to Dropbox

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.

Accessible Recursive Functions
Available formats
×

Save article to Google Drive

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.

Accessible Recursive Functions
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *