# 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 the universe $$V$$ if $$W$$ is a mannequin of ZFC and there’s a poset $$Pin W$$ such that $$W[G]=V$$ for some $$G$$ which is $$P$$-generic over $$W$$. The Ground Axiom (GA) asserts that $$V$$ has no nontrivial grounds whereas 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 GA can breathe pressured (by way of a category forcing understanding) and that DDG is a theorem of ZFC. I’m eager about limiting these notions. For instance, GA$$_{sigma-closed}$$ is the assertion that the universe in not a set-forcing extension of an internal mannequin $$sigma-closed$$ forcing understanding.

Similarly DDG$$_{sigma-closed}$$ 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, GA$$rightarrow$$GA$$_{sigma-closed}$$. How in regards to the discourse? That is, are there fashions of ZFC satisfying $$neg$$GA$$+$$GA$$_{sigma-closed}$$ or $$neg$$GA$$+$$GA$$_{ccc}$$? This query arises from https://arxiv.org/pdf/math/0609270.pdf and when it was printed the query was quiet launch.

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

• DDG$$_{sigma-closed}$$ fails,
• 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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