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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.

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Would you object to the cofinality of $\theta$ being altered? – Noah Schweber May 7 '13 at 20:39
I don't understand the update - won't $V_{\theta+2}$ and $V_\theta$ agree on whether $V_\theta\models ZFC$? – Noah Schweber May 7 '13 at 20:53
Noah, I don't object, not yet. Do you have an idea? – Erin Carmody May 7 '13 at 20:53
@Noah: In general that won't work: We could have $\theta$ measurable, and force it to become of cofinality $\omega$ without adding bounded subsets. Or we could already begin with $\theta$ of cofinality $\omega$, and then there is no cofinality change we can make. – Andrés Caicedo May 7 '13 at 21:21
Here is one way one could make a more definite question: if $\theta$ is worldly, then is there a forcing extension $V[G]$ where $V_\theta^{V[G]}$ satisfies the replacement axiom for $\Sigma_n$ assertions, but not for $\Sigma_{n+1}$ formulas? That is, we want to kill ZFC, but not completely. For which natural numbers $n$ can one do this? – Joel David Hamkins May 7 '13 at 22:05

1 Answer 1

up vote 9 down vote accepted

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.

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I have some thoughts on this, though I don't know if any of them are really that good. First, it is a theorem that set-forcing over a ground model of ZFC will only yield models of ZFC, so starting with $V_\theta$ itself won't work. However, in Joel's response, it appears that we could have started with $V_\theta$ and killed the worldliness of $\theta$. However, the requirement that $\theta$ be singular makes the forcing appear to be a class forcing from the point-of-view of $V_\theta$. If this is the case, then one can't eliminate the hypothesis that $\theta$ be singular here. Is this correct? – Everett Piper May 8 '13 at 3:13
Second, the desired forcing would have to kill some particular axiom of ZFC. Joel suggests altering the truth-value of some particular instance of replacement, but couldn't we attack an instance of collection (I'm not sure if this even makes sense) or even introduce a set into $V_\theta$ that has no choice function, thereby violating choice instead? This seems to require that one look at a symmetric inner model $M$ and inspect its version of $V_\theta$ (though I'm guessing this isn't what you had in mind, Erin). – Everett Piper May 8 '13 at 3:19
As far as the other axioms of ZFC go, you might try to violate them by requiring some kind of definability constraint on $V_\theta$. What I have in mind here is something like: assume $\theta$ is worldly and every set (in $V_\theta$) is definable from parameters in some set $X$ (given ahead of time). Then could there be a forcing extension where, say, pairing fails in the sense that there are sets $a$ and $b$ where $\{a,b\}$ is not definable (from parameters in $X$)? I don't know if this situation could ever arise, but it seems like it could in contexts like $V=L$ or there is an $I_0$ cardinal – Everett Piper May 8 '13 at 3:25
Everett, thanks for your comments. Yes, I am imagining that we are forcing over $V$, and want to kill the worldliness of $\theta$, but softly. So the poset needn't be class forcing in $V_\theta$. In my (new) argument, the forcing is definable in $\langle V_\theta,A,{\in}\rangle$, but this is not a ZFC(A) model, so it isn't really class forcing over $V_\theta$. – Joel David Hamkins May 8 '13 at 3:25
Thank you Joel for your answer and comments! I like your idea on how to stratify worldliness. – Erin Carmody May 8 '13 at 17:34

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