set theory - Which very large cardinals are preserved under Woodin's forcing for $mathsf{AC}$?

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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