Skip to main content Accessibility help
×
Home
Hostname: page-component-768ffcd9cc-96qlp Total loading time: 0.275 Render date: 2022-12-05T23:56:31.943Z Has data issue: true Feature Flags: { "useRatesEcommerce": false } hasContentIssue true

GENERALIZED ALGEBRA-VALUED MODELS OF SET THEORY

Published online by Cambridge University Press:  12 January 2015

BENEDIKT LÖWE*
Affiliation:
Institute for Logic, Language and Computation, Universiteit van Amsterdam and Fachbereich Mathematik, Universität Hamburg
SOURAV TARAFDER*
Affiliation:
Department of Commerce (Morning), St. Xavier’s College and Department of Pure Mathematics, Calcutta University
*
*INSTITUTE FOR LOGIC, LANGUAGE AND COMPUTATION, UNIVERSITEIT VAN AMSTERDAM, POSTBUS 94242, 1090 GE AMSTERDAM, THE NETHERLANDS E-mail: b.loewe@uva.nl, FACHBEREICH MATHEMATIK, UNIVERSITÄT HAMBURG, BUNDESSTRASSE 55, 20146 HAMBURG, GERMANY
DEPARTMENT OF COMMERCE (MORNING), ST. XAVIER’S COLLEGE, 30 MOTHER TERESA SARANI, KOLKATA, 700016, INDIA E-mail: souravt09@gmail.com, DEPARTMENT OF PURE MATHEMATICS, UNIVERSITY OF CALCUTTA, 35 BALLYGUNGE CIRCULAR ROAD, KOLKATA, 700019, INDIA

Abstract

We generalize the construction of lattice-valued models of set theory due to Takeuti, Titani, Kozawa and Ozawa to a wider class of algebras and show that this yields a model of a paraconsistent logic that validates all axioms of the negation-free fragment of Zermelo-Fraenkel set theory.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2014 

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

BIBLIOGRAPHY

Bell, J. L. (2005). Set Theory, Boolean-Valued Models and Independence Proofs (third edition). Oxford Logic Guides, Vol. 47. Oxford: Oxford University Press.CrossRefGoogle Scholar
Brady, R. (1971). The consistency of the axioms of abstraction and extensionality in a three valued logic. Notre Dame Journal of Formal Logic, 12, 447453.CrossRefGoogle Scholar
Brady, R., & Routley, R. (1989). The non-triviality of extensional dialectical set theory. In Priest, G., Routley, R., and Norman, J., editors. Paraconsistent Logic: Essays on the Inconsistent. Analytica. Munich: Philosophia Verlag, pp. 415436.Google Scholar
Carnielli, W. A., & Marcos, J. (2002). A taxonomy of C-systems. In Carnielli, W. A., Coniglio, M. E., and D’Ottaviano, I. M. L., editors. Paraconsistency: The Logical Way to the Inconsistent. Proceedings of the 2nd World Congress on Paraconsistency (WCP 2000). Lecture Notes in Pure and Applied Mathematics, Vol. 228. New York: Marcel Dekker, pp. 194.CrossRefGoogle Scholar
Coniglio, M. E., & da Cruz Silvestrini, L. H. (2014). An alternative approach for quasi-truth. Logic Journal of the IGPL, 22(2), 387410.CrossRefGoogle Scholar
Grayson, R. J. (1979). Heyting-valued models for intuitionistic set theory. In Fourman, M. P., Mulvey, C. J., & Scott, D. S., editors. Applications of Sheaves, Proceedings of the Research Symposium on Applications of Sheaf Theory to Logic, Algebra and Analysis held at the University of Durham, Durham, July 9–21, 1977, Lecture Notes in Mathematics, Vol. 753. Berlin: Springer, pp. 402414.Google Scholar
Libert, T. (2005). Models for paraconsistent set theory. Journal of Applied Logic, 3(1), 1541.CrossRefGoogle Scholar
Marcos, J. (2000). 8k solutions and semi-solutions to a problem of da Costa. Unpublished.
Marcos, J. (2005). On a problem of da costa. In Sica, G., editor. Essays on the Foundations of Mathematics and Logic, Advanced Studies in Mathematics and Logic, Vol. 2. Monza: Polimetrica, pp. 3955.Google Scholar
Ozawa, M. (2007). Transfer principle in quantum set theory. Journal of Symbolic Logic, 72(2), 625648.CrossRefGoogle Scholar
Ozawa, M. (2009). Orthomodular-valued models for quantum set theory. Preprint, arXiv 0908.0367.
Restall, G. (1992). A note on naïve set theory in LP. Notre Dame Journal of Formal Logic, 33(3), 422432.CrossRefGoogle Scholar
Takeuti, G., & Titani, S. (1992). Fuzzy logic and fuzzy set theory. Archive for Mathematical Logic, 32(1), 132.CrossRefGoogle Scholar
Tarafder, S. (2015). Ordinals in an algebra-valued model of a paraconsistent set theory. In Banerjee, M. and Krishna, S., editors. Logic and Its Applications, 6th International Conference, ICLA 2015, Mumbai, India, January 8–10, 2015, Proceedings, Lecture Notes in Computer Science, Vol. 8923. Berlin: Springer-Verlag, pp. 15206.Google Scholar
Titani, S. (1999). A lattice-valued set theory. Archive for Mathematical Logic, 38(6), 395421.CrossRefGoogle Scholar
Titani, S., & Kozawa, H. (2003). Quantum set theory. International Journal of Theoretical Physics, 42(11), 25752602.CrossRefGoogle Scholar
Weber, Z. (2010a). Extensionality and restriction in naive set theory. Studia Logica, 94(1), 87104.CrossRefGoogle Scholar
Weber, Z. (2010b). Transfinite numbers in paraconsistent set theory. Review of Symbolic Logic, 3(1), 7192.CrossRefGoogle Scholar
Weber, Z. (2013). Notes on inconsistent set theory. In Tanaka, K., Berto, F., Mares, E., and Paoli, F., editors. Paraconsistency: Logic and Applications, Logic, Epistemology, and the Unity of Science, Vol. 26. Dordrecht: Springer-Verlag, pp. 315328.CrossRefGoogle Scholar
15
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.

GENERALIZED ALGEBRA-VALUED MODELS OF SET THEORY
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.

GENERALIZED ALGEBRA-VALUED MODELS OF SET THEORY
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.

GENERALIZED ALGEBRA-VALUED MODELS OF SET THEORY
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? *