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NON-CLASSICAL FOUNDATIONS OF SET THEORY

Part of: Set theory Model theory

Published online by Cambridge University Press:  02 December 2021

SOURAV TARAFDER
SOURAV TARAFDER*
Affiliation:
BUSINESS MATHEMATICS AND STATISTICS ST. XAVIER’S COLLEGE 30 MOTHER TERESA SARANI KOLKATA, WEST BENGAL700016, INDIA and INSTITUTO DE FILOSOFIA E CIÊNCIAS HUMANAS UNIVERSITY OF CAMPINAS (UNICAMP) BARÃO GERALDO, SP 13083-896, BRAZILE-mail:[email protected]
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Abstract

In this paper, we use algebra-valued models to study cardinal numbers in a class of non-classical set theories. The algebra-valued models of these non-classical set theories validate the Axiom of Choice, if the ground model validates it. Though the models are non-classical, the foundations of cardinal numbers in these models are similar to those in classical set theory. For example, we show that mathematical induction, Cantor’s theorem, and the Schröder–Bernstein theorem hold in these models. We also study a few basic properties of cardinal arithmetic. In addition, the generalized continuum hypothesis is proved to be independent of these non-classical set theories.

Keywords

algebra-valued modelsnon-classical set theoriescardinalityAxiom of Choicegeneralized continuum hypothesis

MSC classification

Primary: 03C90: Nonclassical models (Boolean-valued, sheaf, etc.)
Secondary: 03E10: Ordinal and cardinal numbers 03E25: Axiom of choice and related propositions 03E50: Continuum hypothesis and Martin's axiom 03E70: Nonclassical and second-order set theories
Type
Article
Information
The Journal of Symbolic Logic , Volume 87 , Issue 1 , March 2022 , pp. 347 - 376
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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References

Aczel, P. and Rathjen, M., Notes on constructive set theory, Report No. 40, 2000/2001, Institut Mittag-Leffler, The Royal Swedish Academy of Sciences, Djursholm, 2001.Google Scholar
Bell, J. L., Set Theory, Boolean-Valued Models and Independence Proofs, third ed., Oxford Logic Guides, vol. 47, The Clarendon Press and Oxford University Press, Oxford, 2005.CrossRefGoogle Scholar
da Costa, N. C. A., Krause, D., and Bueno, O., Paraconsistent logics and paraconsistency , Handbook of the Philosophy of Science, Philosophy of Logic(D. Jacquette, editor) Elsevier, Amsterdam, 2007, pp. 791–911.CrossRefGoogle Scholar
Esser, O., A strong model of paraconsistent logic . Notre Dame Journal of Formal Logic, vol. 44 (2003), no. 3, pp. 149–156.CrossRefGoogle Scholar
Friedman, H. and Ščedrov, A., Large sets in intuitionistic set theory . Annals of Pure and Applied Logic, vol. 27 (1984), no. 1, pp. 1–24.CrossRefGoogle Scholar
Grayson, R. J., Heyting-valued models for intuitionistic set theory , Applications of Sheaves (M. P. Fourman, C. J. Mulvey, and D. S. Scott, editors), Lecture Notes in Mathematics, vol. 753, Springer, Berlin, 1979, pp. 402–414, 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.CrossRefGoogle Scholar
Jockwich, S. and Venturi, G., Non-classical models of ZF . Studia Logica, vol. 109 (2021), pp. 509–537.CrossRefGoogle Scholar
Libert, T., Models for paraconsistent set theory . Journal of Applied Logic, vol. 3 (2005), no. 1, pp. 15–41.CrossRefGoogle Scholar
Löwe, B. and Tarafder, S., The generalized algebra-valued models of set theory . The Review of Symbolic Logic, vol. 8 (2015), no. 1, pp. 192–205.CrossRefGoogle Scholar
Priest, G., Paraconsistent logic , Handbook of Philosophical Logic, vol. 6 (Gabbay, D. and Guenthner, F., editors), Kluwer Academic, Dordrecht, 2002, pp. 287–393.CrossRefGoogle Scholar
Tarafder, S., Ordinals in an algebra-valued model of a paraconsistent set theory , Logic and Its Applications (Banerjee, M. and Krishna, S., editors), Lecture Notes in Computer Science, vol. 8923, Springer, Berlin, 2015, pp. 195–206, Proceedings of the 6th International Conference, ICLA 2015, Mumbai, India, January 8–10, 2015.CrossRefGoogle Scholar
Tarafder, S. and Chakraborty, M. K., A paraconsistent logic obtained from an algebra-valued model of set theory , New Directions in Paraconsistent Logic (Beziau, J. Y., Chakraborty, M. K., and Dutta, S., editors), Springer Proceedings in Mathematics & Statistics, vol. 152, Springer, New Delhi, 2015, pp. 165–183, Proceedings of the 5th WCP, Kolkata, India, February 2014.CrossRefGoogle Scholar
Tarafder, S. and Venturi, G., Independence proofs in non-classical set theories . The Review of Symbolic Logic (2021). doi:10.1017/S1755020321000095.Google Scholar
Weber, Z., Transfinite cardinals in paraconsistent set theory . The Review of Symbolic Logic, vol. 5 (2012), no. 2, pp. 269–293.CrossRefGoogle Scholar