Forcing mildly over a worldly cardinal.

Question

A cardinal $\theta$ is worldly if $V_{\theta}$ is a model of ZFC. We could force to collapse $\theta$ to a successor cardinal, for example, and destroy the worldliness of $\theta$, but is there a less catastrophic way to do so? I'd like to know about a notion of forcing which makes $V_{\theta}$ no longer a model of ZFC in the extension, but in the mildest way possible. Ideally, I'd like $\theta$ in the extension to be very similar to the original $\theta$. In other words, and I'm pretty what I want is not possible, is there a way to force to change $V_{\theta}$ as much as possible while changing $\theta$ as little as possible? Is there any way to tease these two apart? I'm sorry that I'm not being very specific about what I'd like to preserve about $\theta$, but if anyone has any ideas based on what I described, I would appreciate your answer.

Answer 1

I've got it! We can kill the worldliness of a singular worldly cardinals as softly as we like.

Theorem. If $\theta$ is any singular worldly cardinal, then for any natural number $n$ there is a forcing extension $V[G]$ in which $\theta$ remains $\Sigma_n$ worldly, but not worldly, meaning that $V_\theta^{V[G]}$ satisfies the $\Sigma_n$ fragment of ZFC, but not ZFC itself.

Thus, such worldly cardinals can be killed as softly as desired.

Proof. First, we may assume without loss that the GCH holds, by forcing it if necessary. Also, by forcing to collapse the cofinality of $\theta$ to $\omega$, which is small forcing with respect to $\theta$ and therefore preserves the worldliness of $\theta$, we may assume that $\theta$ has cofinality $\omega$.

I claim that in $V$, we may find a set $A\subset\theta$ that is $V_\theta$-generic for the class forcing $\text{Add}(\text{Ord},1)$ to add a Cohen subset of the ordinals over $V_\theta$. To see this, one simply finds ordinals $\theta_n$ with supremum $\theta$ such that $V_{\theta_n}\prec_{\Sigma_n} V_\theta$, and then diagonalizes with respect to the $\Sigma_n$-definable dense classes having parameters in $V_{\theta_n}$ when extending $A$ up to $\theta_{n+1}$. Even though the forcing is not even countably closed (since $\theta$ has cofinality $\omega$), nevertheless we can meet the dense class before the next higher reflecting cardinal since we've limited the complexity of the dense class. It follows that $\langle V_\theta,A,{\in}\rangle$ satisfies $\text{ZFC}(A)$, the theory of ZFC in which the class $A$ is allowed to appear as a predicate the in the replacement scheme.

Now let $\mathbb{Q}$ be the class forcing over $V_\theta$ to code $A$ into the GCH pattern. If $G\subset\mathbb{Q}$ is $V$-generic, then it follows that $V_\theta^{V[G]}=V_\theta[G]$ is a model of ZFC, and so $\theta$ is still worldly in $V[G]$.

But let me now modify the argument slightly, so as to preserve only some amount of worldliness, while killing the rest. The idea is to find a set $A$ in $V$ that is $\Sigma_k$-generic over $V_\theta$, but not fully generic for the definable dense classes in the first step, where $k$ is much larger than $n$. We can ensure that $\langle V_\theta,A,{\in}\rangle$ satisfies the $\Sigma_k$ fragment of $\text{ZFC}(A)$, but not all of $\text{ZFC}(A)$. This can be done by inserting coding information to reveal an unbounded $\omega$-sequence when restricted to the $\Sigma_{k+1}$ reflecting cardinals. In essence, one hides away the cofinal $\omega$-sequence within the complex set of $\Sigma_{k+1}$-reflecting cardinals. A very similar idea is used in the the final section of our paper J. D. Hamkins, D. Linetsky, J. Reitz, Pointwise definable models of set theory.

The point now is that if $k$ is sufficiently larger than $n$, then the $\Sigma_k$ genericity of $A$ will ensure that after one codes $A$ into the GCH pattern of $V[G]$, one still gets that $V_\theta^{V[G]}=V_\theta[G]$ will satisfy at least the $\Sigma_n$ fragment of ZFC. But it will not satisfy all of ZFC, because $A$ is definable in this model and $A$ reveals the unbounded $\omega$-sequence of ordinals. So in $V[G]$, the ordinal $\theta$ is $\Sigma_n$-worldly, but not worldly. QED

As observed earlier, we can extend this result to regular $\theta$ in the case that $\theta$ is measurable, simply by first performing Prikry forcing to singularie $\theta$ while preserving its worldliness, thereby reducing to the singular case above.

Update. But in general, we cannot get the result for all regular worldly cardinals, because if the result holds for a regular worldly cardinal $\theta$, then in fact $\theta$ must be measurable in an inner model. To see this, suppose that $\theta$ is a regular worldly cardinal, which is another way of saying that $\theta$ is inaccessible, and suppose that the conclusion of the result is true for $\theta$. It follows that there is a forcing extension in which $\theta$ is a strong limit cardinal but not worldly, and so in particular $\theta$ is not inaccessible, and thus it is singular in $V[G]$. In other words, we have a forcing extension $V[G]$ in which $\theta$ is a singular cardinal. But this implies by a covering lemma argument with the Dodd-Jensen core model (recently explained to me by Gunter Fuchs) that $\theta$ is measurable in an inner model. So we cannot expect to kill inaccessibility softly down to worldly non-inaccessbility for all inaccessible cardinals.