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

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).

The arguments are nice and short. Thank you for posting them.

Joel, these are really very pretty arguments! Indestructible strongly unfoldable cardinals also have the property that $V_\kappa$ satisfies CCA right?

Thanks very much! And yes, essentially the same argument works with strongly unfoldable. All you need is indestructible $\Sigma_2$-extendible, which is really very weak, weaker than $\Sigma_2$-reflecting, since you don’t need to reflect arbitrarily high, but only a little, since the coding can be done right away above $\kappa$.

That’s nice!

Now I know that indestructible supercompact cardinals have GCH fail as weirdly as possible below them. (So in particular if we assume that $\kappa$ is supercompact and GCH holds, it is not indestructible!)

Thanks!

That’s right, but meanwhile there is an analogue of the Laver preparation that makes the supercompactness of $\kappa$ indestructible by all $\lt\kappa$-directed closed GCH-preserving forcing, with full GCH. See my paper “Destruction or preservation as you like it” http://jdh.hamkins.org/asyoulikeit/.

Joel, that’s an interesting situation then. We can have either “everything” in the continuum below $\kappa$, or nothing at all. Is it also generalizable to fix any pattern of the continuum and then the indestructibility is with regards to $<\kappa$-directed closed forcings which preserve these patterns everywhere.

Or something like that…

That’s right, and that is what the “As you like it” phrase was meant to evoke. You can make the supercompactness of kappa indestructible by whatever class of directed closed you can define in a local manner, and this also allows you to cause certain patterns in the GCH holding or failing, etc.