Infinite Forcing and the Generic Multiverse

Studia Logica - Tập 108 - Trang 277-290 - 2019
Giorgio Venturi1
1Philosophy department Unicamp, Barão Geraldo, Campinas, Brazil

Tóm tắt

In this article we present a technique for selecting models of set theory that are complete in a model-theoretic sense. Specifically, we will apply Robinson infinite forcing to the collections of models of ZFC obtained by Cohen forcing. This technique will be used to suggest a unified perspective on generic absoluteness principles.

Tài liệu tham khảo

Arrigoni, T., and S.D. Friedman, The hyper universe program, Bulletin of Symbolic Logic 19(1):77–96, 2013. Bagaria, J., Bounded forcing axioms as principles of generic absoluteness, Archive for Mathematical Logic 39(6):393–401, 2000. Balaguer, M., Platonism and Anti-platonism in Mathematics, Oxford University Press, 1998. Cherlin, G., A New Approach to the Theory of Infinitely Generic Structures, Dissertation, Yale University, 1971. Cohen, P., Skolem and pessimism about proof in mathematics, Philosophical Transaction of the Royal Society A 363:2407–2418, 2005. Fuchs, G., J.D. Hamkins, and J. Reitz, Set-theoretic geology, Annals of Pure and Applied Logic 166(4):464–501, 2015. Hamkins, J.D., A simple maximality principle, Journal of Symbolic Logic 68(2):527–550, 2003. Hamkins, J.D., The set-theoretic multiverse, Review of Symbolic Logic 5:416–449, 2012. Hamkins, J.D., Upward closure and amalgamation in the generic multiverse of a countable model of set theory, RIMS Kyôkyûroku, pp. 17–31, 2016. Hamkins, J.D., and T. Johnstone, Resurrection axioms and uplifting cardinals, Archive for Mathematical Logic 50(3–4):463–485, 2014. Hirschfeld, J., and W.H. Wheeler, Forcing, Arithmetic, Division Rings, Springer, 1975. Hodges, W., Building Models by Games, Cambridge University Press, 1985. Magidor, M., Some set theories are more equal then others, First draft from the EFI Project website, 2012. Manevitz, L., Robinson forcing is not absolute, Israel Journal of Mathematics 25:211–232, 1976. Väänänen, J., Multiverse set theory and absolutely undecidable propositions, in J. Kennedy (ed.), Interpreting Gödel, Cambridge University Press, 2014, pp. 180–208. Viale, M., Category forcings, MM\(^{+++}\), and generic absoluteness for the theory of strong forcing axioms, Journal of the American Mathematical Society 29(3):675–728, 2016. Woodin, H., The axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal, Walter de Gruyter and Co., 1999. Woodin, H., The Realm of the Infinite, in M. Heller, and H. Woodin (eds.) Infinity. New Research Frontiers, Cambridge University Press, 2011, pp. 89–118.