Assume $(\kappa_n| n<\omega)$ is an increasing sequence of inaccessible cardinals with $\kappa_\omega=\sup_{n<\omega}\kappa_n.$. Let $((\mathbb{P}_n| n \leq \omega), (\dot{\mathbb{Q}}_n | n<\omega))$ be a full support iteration of forcing notions, where for each $n< \omega$, we have $\Vdash_{\mathbb{P}_n}$``$\dot{\mathbb{Q}}_n$ has size < $\kappa_{n+1}$ and does not change $\dot{V}_{\kappa_n}$''.

Question (a) Does forcing with $\mathbb{P}_\omega$ add a new real$?$

 

(b) Does forcing with $\mathbb{P}_\omega$ collapse $\kappa_\omega?$

Remark. I assume each $\kappa_n$ is inaccessible but not Mahlo.

Assume $(\kappa_n| n<\omega)$ is an increasing sequence of inaccessible cardinals with $\kappa_\omega=\sup_{n<\omega}\kappa_n.$. Let $((\mathbb{P}_n| n \leq \omega), (\dot{\mathbb{Q}}_n | n<\omega))$ be a full support iteration of forcing notions, where for each $n< \omega$, we have $\Vdash_{\mathbb{P}_n}$``$\dot{\mathbb{Q}}_n$ has size < $\kappa_{n+1}$ and does not change $\dot{V}_{\kappa_n}$''.

Question (a) Does forcing with $\mathbb{P}_\omega$ add a new real$?$

(b) Does forcing with $\mathbb{P}_\omega$ collapse $\kappa_\omega?$

Remark. I assume each $\kappa_n$ is inaccessible but not Mahlo.

Assume $(\kappa_n| n<\omega)$ is an increasing sequence of inaccessible cardinals with $\kappa_\omega=\sup_{n<\omega}\kappa_n.$. Let $((\mathbb{P}_n| n \leq \omega), (\dot{\mathbb{Q}}_n | n<\omega))$ be a full support iteration of forcing notions, where for each $n< \omega$, we have $\Vdash_{\mathbb{P}_n}$``$\dot{\mathbb{Q}}_n$ has size < $\kappa_{n+1}$ and does not change $\dot{V}_{\kappa_n}$''.

Question (a) Does forcing with $\mathbb{P}_\omega$ add a new real$?$

(b) Does forcing with $\mathbb{P}_\omega$ collapse $\kappa_\omega?$

Assume $(\kappa_n| n<\omega)$ is an increasing sequence of inaccessible cardinals with $\kappa_\omega=\sup_{n<\omega}\kappa_n.$. Let $((\mathbb{P}_n| n \leq \omega), (\dot{\mathbb{Q}}_n | n<\omega))$ be a full support iteration of forcing notions, where for each $n< \omega$, we have $\Vdash_{\mathbb{P}_n}$``$\dot{\mathbb{Q}}_n$ has size < $\kappa_{n+1}$ and does not change $\dot{V}_{\kappa_n}$''.

Question (a) Does forcing with $\mathbb{P}_\omega$ add a new real$?$

(b) Does forcing with $\mathbb{P}_\omega$ collapse $\kappa_\omega?$

Remark. I assume each $\kappa_n$ is inaccessible but not Mahlo.