# clique concept – Restricted notions of set-theoretic geology Answer

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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

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