Some Second Order Set Theory
This article surveys two recent developments in set theory sharing an essential second-order nature, namely, the modal logic of forcing, oriented upward from the universe of set theory to its forcing extensions; and set-theoretic geology, oriented downward from the universe to the inner models over which it arises by forcing. The research is a mixture of ideas from several parts of logic, including, of course, set theory and forcing, but also modal logic, finite combinatorics and the philosophy of mathematics, for it invites a mathematical engagement with various philosophical views on the nature of mathematical existence.
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- 1.Fuchs, G., Hamkins, J.D., Reitz, J.: Set-theoretic geology (in preparation)Google Scholar
- 6.Hamkins, J.D., Seabold, D.: Boolean ultrapowers (in preparation)Google Scholar
- 8.Reitz, J.: The Ground Axiom. Ph.D thesis, The Graduate Center of the City University of New York (September 2006)Google Scholar
- 10.Stavi, J., Väänänen, J.: Reflection principles for the continuum. In: Zhang, Y. (ed.) Logic and Algebra. AMS Contemporary Mathematics Series, vol. 302 (2003)Google Scholar
- 11.Hugh Woodin, W.: Recent development’s on Cantor’s Continuum Hypothesis. In: Proceedings of the Continuum in Philosophy and Mathematics. Carlsberg Academy, Copenhagen (November 2004)Google Scholar