Let $\kappa$ be a regular cardinal, and let $\mathbb{P} = Add(\kappa,1)$ be the standard forcing notion for adding a new subset of $\kappa$ using partial function from $\kappa$ to $2$ with domain of size $<\kappa$. This forcing is $\kappa$-closed, so it doesn't collapse cardinals $\leq \kappa$. If we further assume that $2^{<\kappa} = \kappa$ then it is also $\kappa^+$.c.c., so it doesn't collapse cardinals at all.

If we extend the universe and add new bounded subsets of $\kappa$ then $\mathbb{P}$ doesn't stay $\kappa$-closed, but it might still be $\kappa$-distributive. For example it will be the case when we extend the universe by using a $\kappa$.c.c. forcing, by a theorem of Easton.

The only way that I know to force that $\mathbb{P}$ collapses cardinals is by changing the cofinality of $\kappa$. Indeed, if $\mathbb{Q}$ is a forcing such that $V^{\mathbb{Q}} \models\text{cf }\kappa = \omega < \kappa$, then any $V^{\mathbb{Q}}$-generic filter for $\mathbb{P}$ codes an enumeration of all ordinals below $\kappa$ of order type $\omega$ (take $\{\alpha_n \mid n < \omega\}$ a cofinal sequence at $\kappa$. For every $p \in \mathbb{P}$, there is $n < \omega$ such that $\text{supp }p \subseteq \alpha_n$. So we can extend $p$ to a condition $q$ in which the first $\gamma$ coordinates after $\alpha_n$ are zeros and $q(\alpha_n + \gamma) = 1$. By density arguments this defines an enumeration of all $\kappa$).

Question: Is it consistent that there is regular cardinal $\kappa$ and a forcing $\mathbb{Q}$ that preserves the regularity of $\kappa$ such that $\Vdash_\mathbb{Q} \check{\mathbb{P}}$ collapses $\kappa$?

Question: Is it consistent that for some regular uncountable cardinal $\kappa$ there is no such $\mathbb{Q}$?


1 Answer 1


The following theorem of Stanley "forcing disabled" answers your first question:

Theorem. Assume there is a proper class of weakly compact cardinals. Then there is a class generic extension $V$ of $L$ such that if $P\in L$ is non-trivial and uniform, $\beta$ is the least cardinal such that forcing with $P$ over $L$ adds a new subset of $\beta,$ and $card^V(\beta)$ is not in $V$ successor of a singular cardinal, then forcing with $P$ over $V$ collapses $card^V(\beta)$, if it is $>\omega.$

In fact the above theorem is true, if instead of $L$ we start with any model of $ZFC+GCH$ which has no inner model with a measurable cardinal.

  • 2
    $\begingroup$ In this paper, Stanley also shows that if $V\models CH$ then one can force $(2^{<\omega_1})^V$ to be a special tree (in the generalized meaning) in $V[G]$, without collapsing $\omega_1$. This implies that forcing with $Add(\omega_1,1)^V$ over $V[G]$ collapses $\omega_1$. This result doesn't require any large cardinals. It is not known if it's possible to achieve similar result for regular cardinals above $\omega_1$ without large cardinals. $\endgroup$
    – Yair Hayut
    Oct 17, 2014 at 13:29

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