# Large cardinal indestructibility: two slick new proofs of prior results

$\newcommand\HOD{\text{HOD}}$

I’ve recently found two slick new proofs of some of my prior results on indestructibility, using the idea of an observation of Arthur Apter’s.  What he had noted is:

Observation. (Apter [1])  If $\kappa$ is a Laver indestructible supercompact cardinal, then $V_\kappa\subset\HOD$.  Indeed, $V_\kappa$ satisfies the continuum coding axiom CCA.

Proof. The continuum coding axiom asserts that every set of ordinals is coded into the GCH pattern (it follows that they are each coded unboundedly often). If $x\subset\kappa$ is any bounded set of ordinals, then let $\mathbb{Q}$ be the forcing to code $x$ into the GCH pattern at regular cardinals directly above $\kappa$. This forcing is ${\lt}\kappa$-directed closed, and so by our assumption, $\kappa$ remains supercompact and in particular $\Sigma_2$-reflecting in the extension $V[G]$. Since $x$ is coded into the GCH pattern of $V[G]$, it follows by reflection that $V_\kappa=V[G]_\kappa$ must also think that $x$ is coded, and so $V_\kappa\models\text{CCA}$. QED

First, what I noticed is that this immediately implies that small forcing ruins indestructibility:

Theorem. (Hamkins, Shelah [2], Hamkins [3]) After any nontrivial forcing of size less than $\kappa$, the cardinal $\kappa$ is no longer indestructibly supercompact, nor even indestructibly $\Sigma_2$-reflecting.

Proof.  Nontrivial small forcing $V[g]$ will add a new set of ordinals below $\kappa$, which will not be coded unboundedly often into the continuum function of $V[g]$, and so $V[g]_\kappa$ will not satisfy the CCA.  Hence, $\kappa$ will not be indestructibly $\Sigma_2$-reflecting there. QED

This argument can be seen as essentially related to Shelah’s 1998 argument, given in [2].

Second, I also noticed that a similar idea can be used to prove:

Theorem. (Bagaria, Hamkins, Tsaprounis, Usuba [4])  Superstrong and other large cardinals are never Laver indestructible.

Proof.  Suppose the superstrongness of $\kappa$ is indestructible. It follows by the observation that $V_\kappa$ satisfies the continuum coding axiom. Now force to add a $V$-generic Cohen subset $G\subset\kappa$.  If $\kappa$ were superstrong in $V[G]$, then there would be $j:V[G]\to M$ with $V[G]_{j(\kappa)}=M_{j(\kappa)}$. Since $G$ is not coded into the continuum function, $M_{j(\kappa)}$ does not satisfy the CCA.  This contradicts the elementarity $V_\kappa=V[G]_\kappa\prec M_{j(\kappa)}$. QED

The argument shows that even the $\Sigma_3$-extendibility of $\kappa$ is never Laver indestructible.

I would note, however, that the slick proof does not achieve the stronger result of [4], which is that superstrongness is never indestructible even by $\text{Add}(\kappa,1)$, and that after forcing to add a Cohen subset to $\kappa$ (among any of many other common forcing notions), the cardinal $\kappa$ is never $\Sigma_3$-extendible (and hence not superstrong, not weakly superstrong, and so on).  The slick proof above uses indestructibility by the coding forcing to get the CCA in $V_\kappa$, and it is not clear how one would argue that way to get these stronger results of [4].

[1] Arthur W. Apter and Shoshana Friedman. HOD-supercompactness, inestructibility, and level-by-level equivalence, to appear in Bulletin of the Polish Academy of Sciences (Mathematics).

[2] Joel David Hamkins, Saharon Shelah, Superdestructibility: A Dual to Laver’s Indestructibility,  J. Symbolic Logic, Volume 63, Issue 2 (1998), 549-554.

[3] Joel David Hamkins, Small forcing makes any cardinal superdestructible, J. Symbolic Logic, 63 (1998).

[4] Joan Bagaria, Joel David Hamkins, Konstantinos Tsaprounis, Toshimichi Usuba, Superstrong and other large cardinals are never Laver indestructible, to appear in the Archive of Math Logic (special issue in memory of Richard Laver).

# Boldface resurrection and the strongly uplifting cardinals, the superstrongly unfoldable cardinals and the almost-hugely unfoldable cardinals, BEST 2014

I will speak at the BEST conference, which is held as a symposium in the much larger 95th Annual Meeting of the American Association for the Advancement of Science, at the University of California at Riverside, June 18-20, 2014.

This talk will be for specialists in the BEST symposium.

Abstract.  I shall introduce several new large cardinal concepts, namely, the strongly uplifting cardinals, the superstrongly unfoldable cardinals and the almost-hugely unfoldable cardinals, and prove their tight connection with one another — actually, they are equivalent! — as well as their equiconsistency with several natural instances of the boldface resurrection axiom, such as the boldface resurrection axiom for proper forcing.  This is joint work with Thomas A. Johnstone.

I am also scheduled to give a plenary General Pubic Lecture, entitled Higher infinity and the foundations of mathematics, as a part of the larger AAAS program, to which the general public is invited.