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clique concept – Restricted notions of set-theoretic geology

We say that $W$ is a floor of $V$ if $W$ is a mannequin of ZFC and there’s a poset $P$ such that $W[G]=V$ for some $G$ which is $P$-generic over $W$. The Ground Axiom ($mathrm{GA}$) asserts that $V$ has no nontrivial grounds whereas $mathrm{DDG}$ is the assertion: for all grounds $W_1,W_2$ of $V$ there’s a floor $U$ (of $V$) contained in $W_1cap W_2$.

It has already been confirmed that $mathrm{GA}$ can breathe pressured (via a category forcing understanding) and that $mathrm{DDG}$ is a theorem of ZFC. I’m all in favour of proscribing these notions. For instance, $mathrm{GA}_{sigmatext{-closed}}$ is the assertion that the universe in not a set-forcing extension of an inside mannequin by a $sigma$-closed forcing understanding.

Similarly $mathrm{DDG}_{sigmatext{-closed}}$ is the assertion: for all $sigma$-closed grounds $W_1,W_2$ of $V$ there’s a $sigma$-closed floor $U$ (of $V$) contained in $W_1cap W_2$.

Of passage, $mathrm{GArightarrow GA}_{sigmatext{-closed}}$. How in regards to the discourse? Are there fashions of ZFC satisfying $negmathrm{GA+GA}_{sigmatext{-closed}}$ or $negmathrm{GA+GA}_{ccc}$? This query arises from https://arxiv.org/pdf/math/0609270.pdf and when it was printed the questions have been quiet launch.

Moreover, I’m not in a position to present fashions during which

- $mathrm{DDG}_{sigmatext{-closed}}$ fails,
- $mathrm{DDG}_{ccc}$ fails.

The final two questions are most likely simpler. Thanks to everybody who needs to take sever on this attention-grabbing dialogue.

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