Abstract
Remarkable cardinals were introduced by Schindler, who showed that the existence of a remarkable cardinal is equiconsistent with the assertion that the theory of \({L(\mathbb R)}\) is absolute for proper forcing (Schindler in Bull Symbolic Logic 6(2):176–184, 2000). Here, we study the indestructibility properties of remarkable cardinals. We show that if κ is remarkable, then there is a forcing extension in which the remarkability of κ becomes indestructible by all <κ-closed ≤κ-distributive forcing and all two-step iterations of the form \({Add(\kappa,\theta)*\dot{\mathbb R}}\), where \({\dot{\mathbb R}}\) is forced to be <κ-closed and ≤κ-distributive. In the process, we introduce the notion of a remarkable Laver function and show that every remarkable cardinal carries such a function. We also show that remarkability is preserved by the canonical forcing of the GCH.
Similar content being viewed by others
References
Bagaria, J., Hamkins, J.D., Tsaprounis, K., Usuba, T.: Superstrong and other large cardinals are never Laver indestructible. To appear in Archive for Mathematical Logic (special issue in honor of Richard Laver)
Cheng, Y., Friedman, S.D., Hamkins, J.D.: Large cardinals need not be large in HOD. Ann. Pure Appl. Logic 166(11), 1186–1198 (2015)
Corazza P.: The wholeness axiom and Laver sequences. Ann. Pure Appl. Logic 105(1–3), 157–260 (2000)
Cummings, J.: Iterated forcing and elementary embeddings. In Handbook of Set Theory, vols. 1, 2, 3, pp. 775–883. Springer, Dordrecht (2010)
Dz̆amonja, M., Hamkins, J.D.: Diamond (on the regulars) can fail at any strongly unfoldable cardinal. Ann. Pure Appl. Logic. 144(1–3), 83–95, December 2006. Conference in honor of sixtieth birthday of James E. Baumgartner
Golshani, M.: Woodin’s surgery method (2015) (preprint)
Gitik M., Shelah S.: On certain indestructibility of strong cardinals and a question of Hajnal. Arch. Math. Logic 28(1), 35–42 (1989)
Gitman V., Welch P.D.: Ramsey-like cardinals II. J. Symb. Logic 76(2), 541–560 (2011)
Hamkins, J.D.: A class of strong diamond principles. Preprint (2002)
Hamkins J.D., Johnstone T.A.: Indestructible strong unfoldability. Notre Dame J. Form. Log. 51(3), 291–321 (2010)
Jech, T.: Set theory. Springer Monographs in Mathematics. Springer, Berlin (2003). The third millennium edition, revised and expanded
Kunen K.: Ultrafilters and independent sets. Trans. Am. Math. Soc. 172, 299–306 (1972)
Laver R.: Making the supercompactness of κ indestructible under κ-directed closed forcing. Israel J. Math. 29(4), 385–388 (1978)
Lévy A., Solovay Robert M.: Measurable cardinals and the continuum hypothesis. Israel J. Math. 5, 234–248 (1967)
Magidor M.: On the role of supercompact and extendible cardinals in logic. Israel J. Math. 10, 147–157 (1971)
Schindler R.-D.: Proper forcing and remarkable cardinals. Bull. Symb. Logic 6(2), 176–184 (2000)
Schindler R.-D.: Proper forcing and remarkable cardinals. II. J. Symb. Logic 66(3), 1481–1492 (2001)
Schindler, R.: Remarkable cardinals. In: Infinity, Computability, and Metamathematics: Festschrift in honour of the 60th birthdays of Peter Koepke and Philip Welch. Series: Tributes. College publications, London, GB (2014)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Cheng, Y., Gitman, V. Indestructibility properties of remarkable cardinals. Arch. Math. Logic 54, 961–984 (2015). https://doi.org/10.1007/s00153-015-0453-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00153-015-0453-8