Skip to main content Accessibility help
×
Home
Hostname: page-component-56f9d74cfd-hg4f7 Total loading time: 0.266 Render date: 2022-06-25T12:35:31.802Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "useNewApi": true }

A dichotomy in classifying quantifiers for finite models

Published online by Cambridge University Press:  12 March 2014

Saharon Shelah
Affiliation:
The Hebrew University of Jerusalem, Einstein Institute of Mathematics, Edmond Safra Campus, Givat Ram, Jerusalem 91904, Israel Department of Mathematics, Hill Center-Busch Campus Rutgers, The State University of New Jersey, 110 Frelinghuysen Road, Piscataway, NJ 08854-8019, USA, E-mail: shelah@math.huji.ac.il
Mor Doron
Affiliation:
The Hebrew University of Jerusalem, Einstein Institute of Mathematics, Edmond Safra Campus, Givat Ram, Jerusalem 91904, Israel Department of Mathematics, Hill Center-Busch Campus Rutgers, The State University of New Jersey, 110 Frelinghuysen Road, Piscataway, NJ 08854-8019, USA, E-mail: shelah@math.huji.ac.il

Abstract

We consider a family of finite universes. The second order existential quantifier Q means for each U Є quantifying over a set of n(ℜ)-place relations isomorphic to a given relation. We define a natural partial order on such quantifiers called interpretability. We show that for every Q, either Q is interpretable by quantifying over subsets of U and one to one functions on U both of bounded order, or the logic L(Q) (first order logic plus the quantifier Q) is undecidable.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2005

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]Baldwin, John T., Definable second order quantifiers, Model theoretic logics (Barwise, J. and Feferman, S., editors), Perspectives in Mathematical Logic, Springer-Verlag, 1985, pp. 446478.Google Scholar
[2]Gaifman, Haim, On local and nonlocal properties, Logic Colloquium '81 (Stern, J., editor). North Holland, 1982, pp. 105135.Google Scholar
[3]Lavrov, I. A., The effective non-separability of the set of identically true formulae and the set of finitly refutable formulae for certain elementary theories, Algebra i Logika, vol. 2 (1963). no. 1, pp. 518, (Russian).Google Scholar
[4]Shelah, Saharon, There are just four second-order quantifiers, Israel Journal of Mathematics, vol. 15 (1973), pp. 282300.CrossRefGoogle Scholar
[5]Shelah, Saharon, Classifying of generalized quantifiers, Around classification theory of models, Lecture Notes in Mathematics, no. 1182, Springer-Verlag, 1986. pp. 146.CrossRefGoogle Scholar
[6]Shelah, Saharon, On quantification with a finite universe, this Journal, vol. 65 (2000), pp. 10551075.Google Scholar

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.

A dichotomy in classifying quantifiers for finite models
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.

A dichotomy in classifying quantifiers for finite models
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.

A dichotomy in classifying quantifiers for finite models
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? *