# rigid collapse to $\aleph_1$

Suppose $\kappa$ is inaccessible (or more). Does there exist a $\kappa$-c.c. partial order $\mathbb P \subseteq V_\kappa$ that forces $\kappa = \aleph_1$, with the following property?-- Whenever $G \subseteq \mathbb P$ is generic over $V$, there does not exist any $H \not= G$ in $V[G]$ which is also $\mathbb P$-generic over $V$.

• I imagine that some Easton iteration which collapses the regular cardinals below $\kappa$ while coding the generic into some definable real might do the trick? Sep 21 '17 at 16:12
• If $\kappa$ is weakly compact, then every real added by $\mathbb P$ exists in an intermediate model by a forcing of size $<\kappa$. Sep 21 '17 at 16:14
• Well. Iterate the whole thing with coding the generic into a real or something? Sep 21 '17 at 16:14
• Sounds promising. I don't know this technique. Sep 21 '17 at 16:17
• Well. I'll definitely drop by the KGRC sometime next week. Sep 21 '17 at 16:37

The answer is positive when $\kappa$ is Mahlo. Please email me for a draft if you are interested.