Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-26T21:05:05.943Z Has data issue: false hasContentIssue false

THE MODAL LOGIC OF INNER MODELS

Published online by Cambridge University Press:  09 March 2016

TANMAY INAMDAR
Affiliation:
SCHOOL OF MATHEMATICS UNIVERSITY OF EAST ANGLIA NORWICH RESEARCH PARK NORWICH NR4 7TJ, UKE-mail: [email protected]
BENEDIKT LÖWE
Affiliation:
INSTITUTE FOR LOGIC, LANGUAGE AND COMPUTATION UNIVERSITEIT VAN AMSTERDAM POSTBUS 94242, 1090 GE AMSTERDAM, THE NETHERLANDS FACHBEREICH MATHEMATIK UNIVERSITÄT HAMBURG BUNDESSTRASSE 55, 20146 HAMBURG, GERMANY CORPUS CHRISTI COLLEGE UNIVERSITY OF CAMBRIDGE TRUMPINGTON STREET CAMBRIDGE, CB2 1 RH, ENGLANDE-mail: [email protected]

Abstract

Using techniques developed by Hamkins, Reitz and the second author, we determine the modal logic of inner models.

Type
Articles
Copyright
Copyright © The Association for Symbolic Logic 2016 

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

REFERENCES

Blackburn, Patrick, de Rijke, Maarten, and Venema, Yde, Modal Logic, Cambridge Tracts in Theoretical Computer Science, vol. 53, Cambridge University Press, Cambridge, 2001.CrossRefGoogle Scholar
Block, Alexander C. and Löwe, Benedikt, Modal Logics and Multiverses, RIMS Kôkyûroku, vol. 1949 (2015), pp. 523.Google Scholar
Chagrov, Alexander and Zakharyaschev, Michael, Modal Logic, Oxford Logic Guides, vol. 35, Clarendon Press, 1997.CrossRefGoogle Scholar
Esakia, Leo and Löwe, Benedikt, Fatal Heyting algebras and forcing persistent sentences. Studia Logica, vol. 100 (2012), no. 12, pp. 163173.Google Scholar
Friedman, Sy, Fuchino, Sakaé, and Sakai, Hiroshi, On the set-generic multiverse, 2012, submitted.Google Scholar
Fuchs, Gunter, Closed maximality principles: implications, separations and combinations, this Journal, vol. 73 (2008), no. 1, pp. 276308.Google Scholar
Fuchs, Gunter, Combined maximality principles up to large cardinals, this Journal, vol. 74 (2009), no. 3, pp. 10151046.Google Scholar
Goranko, Valentin and Otto, Martin, Model theory of modal logic, Handbook of Modal Logic (Blackburn, Patrick, van Benthem, Johan F. A. K., and Wolter, Frank, editors), Studies in Logic and Practical Reasoning, vol. 3, Elsevier, 2007, pp. 249330.Google Scholar
David Hamkins, Joel, A simple maximality principle, this Journal, vol. 68 (2003), no. 2, pp. 527550.Google Scholar
David Hamkins, Joel, Some second order set theory, Logic and Its Applications (Ramanujam, R. and Sarukkai, Sundar, editors), Proceedings of the Third Indian Conference, ICLA 2009, Chennai, India, January 7-11, 2009, Lecture Notes in Computer Science, vol. 5378, Springer, Berlin, 2009, pp. 3650.Google Scholar
David Hamkins, Joel, Leibman, George, and Löwe, Benedikt, Structural connections between a forcing class and its modal logic. Israel Journal of Mathematics, vol. 207 (2015), no. 2, pp. 617651.Google Scholar
David Hamkins, Joel and Löwe, Benedikt, The modal logic of forcing. Transactions of the American Mathematical Society, vol. 360 (2008), no. 4, pp. 17931817.Google Scholar
David Hamkins, Joel and Löwe, Benedikt, Moving up and down in the generic multiverse, Logic and Its Applications, 5th International Conference, ICLA 2013, Chennai, India, January 10–12, 2013, Proceedings (Lodaya, Kamal, editor), Lecture Notes in Computer Science, vol. 7750, Springer-Verlag, 2013, pp. 139147.Google Scholar
David Hamkins, Joel and Hugh Woodin, W., The necessary maximality principle for c.c.c. forcing is equiconsistent with a weakly compact cardinal, Mathematical Logic Quarterly, vol. 51 (2005), no. 5, pp. 493498.Google Scholar
Hughes, George E. and Cresswell, Max J., A New Introduction to Modal Logic, Routledge, London, 1996.Google Scholar
Inamdar, Tanmay C., On the Modal Logics of Some Set-Theoretic Constructions, Master’s Thesis Universiteit van Amsterdam, 2013, ILLC Publications MoL-2013-07.Google Scholar
Jech, Thomas, Set Theory, third millennium edition, Perspectives in Mathematical Logic, Springer-Verlag, 2003.Google Scholar
Laver, Richard, Certain very large cardinals are not created in small forcing extensions.. Annals of Pure and Applied Logic, vol. 149 (2007), no 13, pp. 16.Google Scholar
Leibman, George, Consistency Strengths of Modified Maximality Principles, Ph.D. thesis, City University of New York, 2004.Google Scholar
Leibman, George, The consistency strength of MPCCC(ℝ). Notre Dame Journal of Formal Logic, vol. 51 (2010), no. 2, pp. 181193.Google Scholar
McAloon, Kenneth, Consistency results about ordinal definability. Annals of Pure and Applied Logic, vol. 2 (1970/71), no. 4, pp. 449467.Google Scholar
Reitz, Jonas, The Ground Axiom, Ph.D. Thesis, The Graduate Center of the City University of New York, 2006.Google Scholar
Rittberg, Colin Jakob, The Modal Logic of Forcing, Master’s Thesis, Westfälische Wilhelms-Universität Münster, 2010.Google Scholar
Woodin, W. Hugh, Davis, Jacob, and Rodríguez, Daniel, The HOD dichotomy, Appalachian Set Theory: 2006–2012 (Cummings, James and Schimmerling, Ernest, editors), London Mathematical Society Lecture Notes, vol. 406, Cambridge University Press, 2012.Google Scholar