7
$\begingroup$

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 $\mathbb{Q}$?

$\endgroup$
7
$\begingroup$

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.

$\endgroup$
  • 1
    $\begingroup$ 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 '14 at 13:29

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.