Indestructibility properties of remarkable cardinals

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Y. Cheng and V. Gitman, “Indestructibility properties of remarkable cardinals,” Arch. Math. Logic, vol. 54, no. 7-8, pp. 961–984, 2015.

Since the seminal results of Levy and Solovay [1] on the indestructibility of large cardinals by small forcing and Laver on the making a supercompact cardinal indestructible by all $\lt\kappa$-directed closed forcing [2], indestructibility properties of various large cardinal notions have been intensively studied. A decade after Laver's result, Gitik and Shelah showed that strong cardinals can be made indestructible by all weakly $\leq\kappa$-closed forcing with the Prikry property [3] (see this post), a class that includes all $\leq\kappa$-closed forcing, and Woodin showed, using his technique of surgery, that they can be made indestructible by forcing of the form ${\rm Add}(\kappa,\theta)$. More recently, Hamkins and Johnstone showed that strongly unfoldable cardinals can be made indestructible by all $\lt\kappa$-closed $\kappa^+$-preserving forcing [4]. It turns out that not all large cardinals possess robust indestructibility properties. Very recently, Bagaria et al. showed that a number of large cardinal notions including superstrong, huge, and rank-into-rank cardinals are superdestructible: they cannot even be indestructible by ${\rm Add}(\kappa,1)$ [5]. In this article, we show that remarkable cardinals have indestructibility properties resembling those of strong cardinals. They can be made indestructible by all $\lt\kappa$-closed $\leq\kappa$-distributive forcing and by all two-step iterations of the form ${\rm Add}(\kappa,\theta)*\dot{\mathbb R}$, where $\dot{\mathbb R}$ is forced to be $\lt\kappa$-closed and $\leq\kappa$-distributive.

Theorem: If $\kappa$ is remarkable, then there is a forcing extension in which the remarkability of $\kappa$ is becomes indestructible by all $\lt\kappa$-closed $\leq\kappa$-distributive forcing and by all two-step iterations ${\rm Add}(\kappa,\theta)*\dot{\mathbb R}$, where $\dot{\mathbb R}$ is forced to be $\lt\kappa$-closed and $\leq\kappa$-distributive.

In particular, a remarkable $\kappa$ can be made indestructible by all $\leq\kappa$-closed forcing, and since $\dot {\mathbb R}$ can be trivial, by all forcing of the form ${\rm Add}(\kappa,\theta)$. One application of the main theorem is that any ${\rm GCH}$ pattern can be forced above a remarkable cardinal. Another application uses a recent forcing construction of Cheng, Hamkins, and Friedman [6], to produce a remarkable cardinal that is not remarkable in ${\rm HOD}$. Using techniques from the proof the main theorem, we also show that remarkability is preserved by the canonical forcing of the ${\rm GCH}$.

For the indestructibility arguments, we define the notion of a remarkable Laver function and show that every remarkable cardinal carries such a function. Although Laver-like functions can be forced to exist for many large cardinal notions [7], remarkable cardinals along with supercompact and strong cardinals are some of the few that outright possess such fully set-anticipating functions. For instance, not every strongly unfoldable cardinal has a Laver-like function because it is consistent that $\diamondsuit_\kappa({\rm REG})$ fails at a strongly unfoldable cardinal [8].

Definition: A cardinal $\kappa$ is remarkable if in the ${\rm Coll}(\omega,\lt\kappa)$ forcing extension $V[G]$, for every regular cardinal $\lambda>\kappa$, there is a $V$-regular cardinal $\overline\lambda<\kappa$ and $j:H_{\overline\lambda}^V\to H_\lambda^V$ with critical point $\gamma$ such that $j(\gamma)=\kappa$.

Schindler originally used a different primary characterization of remarkable cardinals, but he has recently switched to using the above characterization, which gives remarkable cardinals a character of generic supercompactness [9]. Magidor showed that $\kappa$ is supercompact if and only if for every regular cardinal $\lambda>\kappa$, there is a regular cardinal $\overline \lambda<\kappa$ and $j:H_{\overline\lambda}\to H_\lambda$ with critical point $\gamma$ such $j(\gamma)=\kappa$ [10].

Schindler introduced remarkable cardinals when he isolated them as the large cardinal notion whose existence is equiconsistent with the assertion that the theory of $L(\mathbb R)$ cannot be altered by proper forcing. The assertion that the theory of $L(\mathbb R)$ is absolute for all set forcing is intimately connected with ${\rm AD}^{L(\mathbb R)}$ and its consistency strength lies in the neighborhood of infinitely many Woodin cardinals [11]. In contrast, remarkable cardinals are much weaker than measurable cardinals, and indeed they can exist in $L$. Consistency-wise, they fit tightly into the $\alpha$-iterable hierarchy of large cardinal notions (below a Ramsey cardinal) introduced by Gitman and Welch [12], where they lie above 1-iterable, but below 2-iterable cardinals, placing them above hierarchies of ineffability, but much below an $\omega$-Erdős cardinal. Remarkable cardinals are also totally indescribable and $\Sigma_2$-reflecting. Strong cardinals are remarkable, but the least measurable cardinal cannot be remarkable by $\Sigma_2$-reflection. For more background to remarkable cardinals, see this post.

References

  1. A. Lévy and R. M. Solovay, “Measurable cardinals and the continuum hypothesis,” Israel J. Math., vol. 5, pp. 234–248, 1967.
  2. R. Laver, “Making the supercompactness of $κ$ indestructible under $κ$-directed closed forcing,” Israel J. Math., vol. 29, no. 4, pp. 385–388, 1978.
  3. M. Gitik and S. Shelah, “On certain indestructibility of strong cardinals and a question of Hajnal,” Arch. Math. Logic, vol. 28, no. 1, pp. 35–42, 1989.
  4. J. D. Hamkins and T. A. Johnstone, “Indestructible strong unfoldability,” Notre Dame J. Form. Log., vol. 51, no. 3, pp. 291–321, 2010.
  5. J. Bagaria, J. D. Hamkins, K. Tsaprounis, and T. Usuba, “Superstrong and other large cardinals are never Laver indestructible,” Arch. Math. Logic, vol. 55, no. 1-2, pp. 19–35, 2016.
  6. Y. Cheng, S.-D. Friedman, and J. D. Hamkins, “Large cardinals need not be large in HOD,” Ann. Pure Appl. Logic, vol. 166, no. 11, pp. 1186–1198, 2015.
  7. J. D. Hamkins, “A class of strong diamond principles,” Manuscript, 2002.
  8. Dz̆amonja Mirna and J. D. Hamkins, “Diamond (on the regulars) can fail at any strongly unfoldable cardinal,” Ann. Pure Appl. Logic, vol. 144, no. 1-3, pp. 83–95, Dec. 2006.
  9. R. Schindler, “Remarkable cardinals,” in Infinity, computability, and metamathematics, vol. 23, Coll. Publ., London, 2014, pp. 299–308.
  10. M. Magidor, “On the role of supercompact and extendible cardinals in logic,” Israel J. Math., vol. 10, pp. 147–157, 1971.
  11. R.-D. Schindler, “Proper forcing and remarkable cardinals,” Bull. Symbolic Logic, vol. 6, no. 2, pp. 176–184, 2000.
  12. V. Gitman and P. D. Welch, “Ramsey-like cardinals II,” The Journal of Symbolic Logic, vol. 76, no. 2, pp. 541–560, 2011.