TITLE: A specific Model of ZFC
QUESTION [6 upvotes]: In his paper "Some Second Order Set Theory", Joel Hamkins asked whether there is a model of set theory $V$ that is elementary equivalent to $V[G]$, Whenever $G$ is $V$-generic for the collapse of a cardinal $\delta$ to $\omega$?
It seems that the existence of such a model will have large cardinal strength.
Assume such model exists, then ;

1.There is no switches(a statement $\phi$ in language of set theory such that both $\phi$ and $\neg \phi$ can be forced by set forcing-here by collapse forcing)for collapse forcing over this model.
2. $\diamondsuit\phi \longleftrightarrow \phi \longleftrightarrow\square \phi$(By collapse forcing).
3. $V \models$ MP(Hamkins' maximality principle)

Assume such a model exists, my questions are :

Question 1. What is valid principle for collapse forcing? Is it modal logic of provability?
Question 2. What are other straightforward consequences?
Question 3. In general, is this model strange? Is it ugly or no, it's nice? will we have nice consequences or not(not necessarily related to modal logic of forcing)?
Question 4. Can this question be asked about other forcing notions, I mean are they interesting?

REPLY [12 votes]: In a joint work with Mitchell, which is under preparation, we succeeded to give a full answer to Hamkins-Löwe question.


Edit:
The paper is ready. See
On a Question of Hamkins and Löwe on the modal logic of collapse forcing.
The proof uses Radin forcing with interleaved collapses. Also it is shown (by Mitchell) that some large cardinals are needed to obtain a consistent answer.