Asaf Karagila
I don't have much choice...

I just posted another problem in the problems page. The prize, by the way, is a bottle of port wine, or equivalent. And I truly hope to make good on that prize.

In another problem there, coming from a work with David Asperó, we asked if an $$\omega_2$$-closed forcing must preserve the property of being proper. Yasou Yoshinobu provided us with a negative answer based on Shelah's "Proper and Improper Forcing" XVII Observation 2.12 (p.826). Take $$\kappa$$ to be uncountable, by forcing with $$\Add(\omega,1)\ast\Col(\omega_1,2^\kappa)$$ and appealing to the gap lemma, $$(2^{<\kappa})$$ is a tree with only $$\aleph_1$$ branches. It can therefore be specialized by a ccc forcing in that model. The iteration of these three forcing (Cohen real, collapse, specialize) is clearly proper. But now by forcing with $$\Add(\kappa,1)$$ we must in fact violate the properness of this forcing, which was defined in the ground model, since the new branch is also generic for the tree and will therefore collapse $$\omega_1$$.

Similarly, Shelah gave a solution where a $$\sigma$$-closed forcing is not proper after adding just a Cohen real. You can find the solution, provided kindly by Martin Goldstern, in this MathOverflow answer.

And this leads us to the point where we realize that provably in $$\ZFC$$ that for any $$\kappa$$ there is always a forcing $$\QQ$$ which is proper in $$V$$, but not proper after forcing with $$\Add(\kappa,1)$$. And that leads us to the following question.

Question. If $$\PP$$ is any nontrivial forcing, is there a forcing $$\QQ$$ which is proper in $$V$$, but not proper after forcing with $$\PP$$?

Well, what are you still doing here? Go! Solve! Earn that bottle of port!

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