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A STRONG REFLECTION PRINCIPLE

Published online by Cambridge University Press:  01 November 2017

SAM ROBERTS
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Abstract

This article introduces a new reflection principle. It is based on the idea that whatever is true in all entities of some kind is also true in a set-sized collection of them. Unlike standard reflection principles, it does not re-interpret parameters or predicates. This allows it to be both consistent in all higher-order languages and remarkably strong. For example, I show that in the language of second-order set theory with predicates for a satisfaction relation, it is consistent relative to the existence of a 2-extendible cardinal (Theorem 7.12) and implies the existence of a proper class of 1-extendible cardinals (Theorem 7.9).

Keywords

03A0503E5503E6500A30set theoryreflection principleslarge cardinalsintrinsic justification

Information

Type
Research Article
Information
The Review of Symbolic Logic , Volume 10 , Issue 4 , December 2017 , pp. 651 - 662
Copyright
Copyright © Association for Symbolic Logic 2017 

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References

BIBLIOGRAPHY

Bernays, P. (1976). On the problem of schemata of infinity in axiomatic set theory. In Müller, G. H., editor. Sets and Classes: On the Work by Paul Bernays. Studies in Logic and the Foundations of Mathematics, Vol. 84. North-Holland: Amsterdam, pp. 121–172.Google Scholar
Boolos, G. (1984). To be is to be a value of a variable (or to be some values of some variables). Journal of Philosophy, 81(8), 430–449.Google Scholar
Boolos, G. (1989). Iteration again. Philosophical Topics, 17, 5–21.Google Scholar
Burgess, J. P. (2004). E pluribus unum: Plural logic and set theory. Philosophia Mathematica, 12(3), 193–221.Google Scholar
Foreman, M. & Kanamori, A. (2009). Handbook of Set Theory. Netherlands: Springer.Google Scholar
Hamkins, J., Gitman, V., & Johnstone, T. (2015). Kelley-morse set theory and choice principles for classes. Available at: http://boolesrings.org/victoriagitman/files/2015/01/kelleymorse2.pdf (accessed June 24, 2016).Google Scholar
Kanamori, A. (2003). The Higher Infinite (second edition). Berlin: Springer.Google Scholar
Koellner, P. (2009). On reflection principles. Annals of Pure and Applied Logic, 157(2–3), 206–219.Google Scholar
Linnebo, O. (2007). Burgess on plural logic and set theory. Philosophia Mathematica, 15, 79–93.Google Scholar
Linnebo, O. & Rayo, A. (2012). Hierarchies ontological and ideological. Mind, 121(482), 269–308.CrossRefGoogle Scholar
Marshall R., M. V. (1989). Higher order reflection principles. Journal of Symbolic Logic, 54(2), 474–489.Google Scholar
Reinhardt, W. N. (1974). Remarks on reflection principles, large cardinals, and elementary embeddings. In Jech, T., editor. Axiomatic Set Theory. Providence: American Mathematical Society, pp. 189–205.Google Scholar
Reinhardt, W. N. (1980). Satisfaction definitions and axioms of infinity in a theory of properties with necessity operator. In Mathematical logic in Latin America. Studies in Logic and the Foundations of Mathematics, Vol. 99. Amsterdam: North-Holland, pp. 267–303.Google Scholar
Tait, W. W. (2003). Zermelo’s conception of set theory and reflection principles. In Schirn, M., editor, Philosophy of Mathematics Today. New York: Oxford University Press, pp. 469–483.Google Scholar
Tait, W. (2005). Constructing cardinals from below. In The Provenance of Pure Reason. Logic and Computation in Philosophy. New York: Oxford University Press, pp. 133–154.CrossRefGoogle Scholar
Uzquiano, G. (2003). Plural quantification and classes. Philosophia Mathematica, 11(3), 67–81.Google Scholar
Welch, P. (forthcoming). Global reflection principles. In Sober, E., Niiniluoto, I., and Leitgeb, H., editors. Proceedings of the CLMPS, Helsinki 2015. London: College Publications.Google Scholar
Williamson, T. (2013). Modal Logic as Metaphysics. Oxford: Oxford University Press.Google Scholar