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CAN MODALITIES SAVE NAIVE SET THEORY?

Published online by Cambridge University Press:  21 December 2017

PETER FRITZ*
Affiliation:
IFIKK, University of Oslo
HARVEY LEDERMAN*
Affiliation:
Department of Philosophy, Princeton University
TIANKAI LIU*
Affiliation:
Department of Mathematics, University of Utah
DANA SCOTT*
Affiliation:
Visiting Scholar, University of California, Berkeley
*
*IFIKK UNIVERSITY OF OSLO PO BOX 1020 BLINDERN 0315 OSLO, NORWAY E-mail: peter.fritz@ifikk.uio.no
DEPARTMENT OF PHILOSOPHY PRINCETON UNIVERSITY PRINCETON, NJ 08544, USA E-mail: harvey.lederman@princeton.edu
DEPARTMENT OF MATHEMATICS UNIVERSITY OF UTAH SALT LAKE CITY, UT 84112, USA E-mail: tliu@math.utah.edu
§VISITING SCHOLAR UNIVERSITY OF CALIFORNIA, BERKELEY BERKELEY, CA 94720, USA E-mail: dana.scott@cs.cmu.edu
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Abstract

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To the memory of Prof. Grigori Mints, Stanford University

Born: June 7, 1939, St. Petersburg, Russia

Died: May 29, 2014, Palo Alto, California

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2017 

References

BIBLIOGRAPHY

Aczel, P. & Feferman, S. (1980). Consistency of the unrestricted abstraction principle using an intensional equivalence operator. In Seldin, J. P. and Hindley, J. R., editors. To H. B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism. New York: Academic Press, pp. 6798.Google Scholar
Bacon, A. (2015). Can the classical logician avoid the revenge paradoxes? Philosophical Review, 124(3), 299352.CrossRefGoogle Scholar
Balaguer, M. (2008). Fictionalism in the philosophy of mathematics. In Zalta, E. N., editor. Stanford Encyclopedia of Philosophy. Available at https://plato.stanford.edu/archives/sum2015/entries/fictionalism-mathematics/.Google Scholar
Blass, A. (1990). Infinitary combinatorics and modal logic. The Journal of Symbolic Logic, 55(2), 761778.CrossRefGoogle Scholar
Boolos, G. (1995). The Logic of Provability. Cambridge: Cambridge University Press.Google Scholar
Cantini, A. (2009). Paradoxes, self-reference and truth in the 20th century. In Woods, J. and Gabbay, D. M. editors. Handbook of the History of Logic, Volume 5: Logic from Russell to Church. Amsterdam: Elsevier, pp. 8751013.Google Scholar
Feferman, S. (1984). Toward useful type-free theories, I. The Journal of Symbolic Logic, 49, 75111.CrossRefGoogle Scholar
Field, H. (1980). Science Without Numbers. Princeton, NJ: Princeton University Press.Google Scholar
Field, H. (1989). Realism, Mathematics & Modality. Oxford: Basil Blackwell.Google Scholar
Field, H., Lederman, H., & Øgaard, T. F. (2017). Prospects for a naive theory of classes. Notre Dame Journal of Formal Logic, 58(4), 461506.CrossRefGoogle Scholar
Fine, K. (1978). Model theory for modal logic – Part II. The elimination of de re modality. Journal of Philosophical Logic, 7, 277306.Google Scholar
Fine, K. (1981). First-order modal theories I—sets. Noûs, 15, 177205.CrossRefGoogle Scholar
Fitch, F. B. (1942). A basic logic. The Journal of Symbolic Logic, 7(3), 105114.CrossRefGoogle Scholar
Fitch, F. B. (1948). An extension of basic logic. The Journal of Symbolic Logic, 13(2), 95106.CrossRefGoogle Scholar
Fitch, F. B. (1963). The system CΔ of combinatory logic. The Journal of Symbolic Logic, 28(1), 8797.CrossRefGoogle Scholar
Fitch, F. B. (1966). A consistent modal set theory (Abstract). The Journal of Symbolic Logic, 31, 701.Google Scholar
Fitch, F. B. (1967a). A complete and consistent modal set theory. The Journal of Symbolic Logic, 32, 93103.CrossRefGoogle Scholar
Fitch, F. B. (1967b). A theory of logical essences. The Monist, 51, 104109.CrossRefGoogle Scholar
Fitch, F. B. (1970). Correction to a paper on modal set theory. The Journal of Symbolic Logic, 35, 242.CrossRefGoogle Scholar
Fitting, M. (2003). Intensional logic—beyond first order. In Hendricks, V. F. and Malinowski, J., editors. Trends in Logic. Dordrecht: Springer, pp. 87108.CrossRefGoogle Scholar
Forster, T. E. (1995). Set Theory with a Universal Set (second edition). Oxford: Clarendon Press.CrossRefGoogle Scholar
Fritz, P., & Goodman, J. (2016). Higher-order contingentism, part 1: Closure and generation. Journal of Philosophical Logic, 45(6), 645695.CrossRefGoogle Scholar
Gallin, D. (1975). Intensional and Higher-Order Modal Logic. Amsterdam: North-Holland.Google Scholar
Gilmore, P. C. (1967). The consistency of a positive set theory. Technical Report RC-1754, IBM Research Report.Google Scholar
Goodman, N. (1990). Topological models of epistemic set theory. Annals of Pure and Applied Logic, 46, 147167.CrossRefGoogle Scholar
Hamkins, J. D. (2003). A simple maximality principle. The Journal of Symbolic Logic, 68(2), 527550.CrossRefGoogle Scholar
Hamkins, J. D. & Löwe, B. (2008). The modal logic of forcing. Transactions of the American Mathematical Society, 360(4), 17931817.CrossRefGoogle Scholar
Hamkins, J. D. & Woodin, W H. (2005). The necessary maximality principle for ccc forcing is equiconsistent with a weakly compact cardinal. Mathematical Logic Quarterly, 51(5), 493498.CrossRefGoogle Scholar
Hellman, G. (1989). Mathematics Without Numbers: Towards a Modal-Structural Interpretation. Oxford: Oxford University Press.Google Scholar
Hughes, G. E. & Cresswell, M. J. (1996). A New Introduction to Modal Logic. London: Routledge.CrossRefGoogle Scholar
Kaye, R. (1993). Review of Forster, T. E., set theory with a universal set: Exploring an untyped universe. Notre Dame Journal of Formal Logic, 34, 302309.CrossRefGoogle Scholar
Krajíček, J. (1987). A possible modal formulation of comprehension scheme. Zeitschrift für mathematische Logik und Grundlagen der Mathematik, 33, 461480.CrossRefGoogle Scholar
Krajíček, J. (1988). Some results and problems in the modal set theory MST. Zeitschrift für mathematische Logik und Grundlagen der Mathematik, 34, 123134.CrossRefGoogle Scholar
Kripke, S. A. (1959). A completeness theorem in modal logic. The Journal of Symbolic Logic, 24, 114.CrossRefGoogle Scholar
Kripke, S. A. (1975). Outline of a theory of truth. Journal of Philosophy, 72(19), 690716.CrossRefGoogle Scholar
Linnebo, Ø. (2010). Pluralities and sets. The Journal of Philosophy, 107(3), 144164.CrossRefGoogle Scholar
Linnebo, Ø. (2013). The potential hierarchy of sets. The Review of Symbolic Logic, 6(2), 205228.CrossRefGoogle Scholar
Parsons, C. (1983). Sets and modality. Mathematics in Philosophy. Ithaca: Cornell University Press, pp. 298341.Google Scholar
Rosen, G. (2001). Nominalism, naturalism, epistemic relativism. Noûs, 35(s15), 6991.CrossRefGoogle Scholar
Rundle, B. (1969). Review of Frederic B. Fitch, “A theory of logical essences” and “A complete and consistent modal set theory”. The Journal of Symbolic Logic, 34, 125.CrossRefGoogle Scholar
Scott, D. (2010). A Boolean-Valued Modal Set Theory. Unpublished Oxford Lectures. Available at: www.homepages.inf.ed.ac.uk/als/ScottInScotland/ScottLect3.pdf.Google Scholar
Scroggs, S. J. (1951). Extensions of the Lewis system S5. The Journal of Symbolic Logic, 16(2), 112120.CrossRefGoogle Scholar
Shapiro, S. (editor) (1985). Intensional Mathematics. Amsterdam: North-Holland.Google Scholar
Smullyan, R. & Fitting, M. (1996). Set Theory and the Continuum Problem. Oxford: Oxford University Press.Google Scholar
Solovay, R. M. (1976). Provability interpretations of modal logic. Israel Journal of Mathematics, 25(3–4), 287304.CrossRefGoogle Scholar
Studd, J. (2013). The iterative conception of set. A (bi-)modal axiomatisation. Journal of Philosophical Logic, 42(5), 697725.CrossRefGoogle Scholar
Williamson, T. (2013). Modal Logic as Metaphysics. Oxford: Oxford University Press.CrossRefGoogle Scholar
Yablo, S. (2001). Go figure: A path through fictionalism. Midwest Studies in Philosophy, 25(1), 72102.CrossRefGoogle Scholar
Zalta, E. N. (1988). Intensional Logic and the Metaphysics of Intentionality. Cambridge, MA: MIT Press.Google Scholar