Skip to main content Accessibility help
×
Home
Hostname: page-component-559fc8cf4f-6f8dk Total loading time: 1.366 Render date: 2021-03-06T04:34:25.671Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": false, "newCiteModal": false, "newCitedByModal": true }

Theories very close to PA where Kreisel's Conjecture is false

Published online by Cambridge University Press:  12 March 2014

Pavel Hrubeš
Affiliation:
Mathematical Institute, Academy of Sciences of the Czech Republic, Prague, Czech Republic. E-mail: pahrubes@centrum.cz
Corresponding
E-mail address:

Abstract

We give four examples of theories in which Kreisel's Conjecture is false: (1) the theory PA(-) obtained by adding a function symbol minus, ‘—’, to the language of PA, and the axiom ∀x∀y∀z (xy = z) ≡ (x = y + z ∨ (x < yz = 0)); (2) the theory L of integers; (3) the theory PA(q) obtained by adding a function symbol q (of arity ≥ 1) to PA, assuming nothing about q; (4) the theory PA(N) containing a unary predicate N(x) meaning ‘x is a natural number’. In Section 6 we suggest a counterexample to the so called Sharpened Kreisel's Conjecture.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2007

Access options

Get access to the full version of this content by using one of the access options below.

References

Baaz, M. and Pudlák, P. [1993], Kreisel's conjecture for L∃1, Arithmetic, proof theory, and computation complexity, Papers from the Conference Held in Prague, 07 2–5, 1991, Oxford University Press, New York, pp. 30–60.Google Scholar
Friedman, H. [1975], One hundred and two problems in mathematical logic, this Journal, vol. 40, pp. 113–129.Google Scholar
Krajiček, J. and Pudlák, P. [1988], The number of proof lines and the size of proofs in first order logic, Archive for Mathematical Logic, vol. 27, pp. 69–84.CrossRefGoogle Scholar
Miyatake, T. [1980], On the lengths of proofs in formal systems, Tsukuba Journal of Mathematics, vol. 4, pp. 115–125.CrossRefGoogle Scholar
Parikh, R. [1973], Some results on the length of proofs, Transactions of the American Mathematical Society, vol. 177, pp. 29–36.CrossRefGoogle Scholar
Yukami, T. [1978], A note on a formalized arithmetic with function symbols and +, Tsukuba Journal of Mathematics, vol. 7, pp. 69–73.Google Scholar
Yukami, T. [1984], Some results on speed-up, Annals of the Japan Association for Philosophy and Science, vol. 6, pp. 195–205.CrossRefGoogle Scholar

Full text views

Full text views reflects PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views.

Total number of HTML views: 0
Total number of PDF views: 11 *
View data table for this chart

* Views captured on Cambridge Core between September 2016 - 6th March 2021. This data will be updated every 24 hours.

Send article to Kindle

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 sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent 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.

Theories very close to PA where Kreisel's Conjecture is false
Available formats
×

Send article to Dropbox

To send 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 use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

Theories very close to PA where Kreisel's Conjecture is false
Available formats
×

Send article to Google Drive

To send 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 use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

Theories very close to PA where Kreisel's Conjecture is false
Available formats
×
×

Reply to: Submit a response


Your details


Conflicting interests

Do you have any conflicting interests? *