Abstract
We give a characterization of extendibility in terms of embeddings between the structures H λ . By that means, we show that the GCH can be forced (by a class forcing) while preserving extendible cardinals. As a corollary, we argue that such cardinals cannot in general be made indestructible by (set) forcing, under a wide variety of forcing notions.
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References
Brooke-Taylor, A.: Large Cardinals and L-Like Combinatorics. Ph.D. Dissertation, University of Vienna (2007)
Brooke-Taylor A.: Indestructibility of Vopěnka’s principle. Arch. Math. Logic 50(5), 515–529 (2011)
Brooke-Taylor A., Friedman S.D.: Large cardinals and gap-1 morasses. Ann. Pure Appl. Logic 159(1–2), 71–99 (2009)
Corazza P.: Laver sequences for extendible and super-almost-huge cardinals. J. Symb. Logic 64(3), 963–983 (1999)
Dobrinen N., Friedman S.D.: Homogeneous iteration and measure one covering relative to HOD. Arch. Math. Logic 47(7–8), 711–718 (2008)
Friedman S.: Accessing the switchboard via set forcing. Math. Logic Q. 58(4–5), 303–306 (2012)
Friedman, S.D.: Large cardinals and L-like universes. In: Andretta, A. (ed.) Set Theory: Recent Trends and Applications, Quaderni di Matematica, vol. 17, pp. 93–110. Seconda Università di Napoli (2005)
Gitman, V., Hamkins, J.D., Johnstone, T.: What is the theory ZFC without power set? Preprint (2011). http://arxiv.org/abs/1110.2430
Hamkins J.D.: Fragile measurability. J. Symb. Logic 59(1), 262–282 (1994)
Hamkins J.D.: The lottery preparation. Ann. Pure Appl. Logic 101(2–3), 103–146 (2000)
Hamkins, J.D.: A class of strong diamond principles. Preprint (2002). http://arxiv.org/abs/math/0211419
Jech T.: Set Theory, The Third Millennium Edition. Springer, Berlin (2002)
Jensen, R.B.: Measurable cardinals and the GCH. In: Jech, T. (ed.) Axiomatic Set Theory, Proceedings of Symposia in Pure Mathematics, vol. 13 (II), pp. 175–178. American Mathematical Society (1974)
Kunen, K.: Set theory, an introduction to independence proofs. In: Barwise, J., Keisler, H.J., Suppes, P., Troelstra, A.S. (eds.) Studies in Logic and the Foundations of Mathematics, vol. 102. North-Holland, Amsterdam (1980)
Laver R.: Making the supercompactness of κ indestructible under κ-directed closed forcing. Israel J. Math. 29, 385–388 (1978)
Menas T.K.: Consistency results concerning supercompactness. Trans. Am. Math. Soc. 223, 61–91 (1976)
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Tsaprounis, K. On extendible cardinals and the GCH. Arch. Math. Logic 52, 593–602 (2013). https://doi.org/10.1007/s00153-013-0333-z
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DOI: https://doi.org/10.1007/s00153-013-0333-z