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 $V$.

  • $\begingroup$ 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? $\endgroup$ – Asaf Karagila Sep 21 '17 at 16:12
  • $\begingroup$ If $\kappa$ is weakly compact, then every real added by $\mathbb P$ exists in an intermediate model by a forcing of size $<\kappa$. $\endgroup$ – Monroe Eskew Sep 21 '17 at 16:14
  • $\begingroup$ Well. Iterate the whole thing with coding the generic into a real or something? $\endgroup$ – Asaf Karagila Sep 21 '17 at 16:14
  • $\begingroup$ Sounds promising. I don't know this technique. $\endgroup$ – Monroe Eskew Sep 21 '17 at 16:17
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    $\begingroup$ Well. I'll definitely drop by the KGRC sometime next week. $\endgroup$ – Asaf Karagila Sep 21 '17 at 16:37

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

  • $\begingroup$ I'd be interested to see the write up. However, I don't have a current email for you, complicating things. $\endgroup$ – Not Mike Nov 8 '17 at 18:58

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