We say that $W$ is a ground of the universe $V$ if $W$ is a model of ZFC and there is a poset $P\in W$ such that $W[G]=V$ for some $G$ which is $P$-generic over $W$. The Ground Axiom ($\text{GA}$) asserts that $V$ has no nontrivial grounds while $\text{DDG}$ is the statement: for all grounds $W_1$, $W_2$ of $V$ there is a ground $U$ (of $V$) contained in $W_1\cap W_2$.

It has already been proven that $\text{GA}$ can be forced (by means of a class forcing notion) and that $\text{DDG}$ is a theorem of ZFC. I'm interested in restricting these notions. For example, $\text{GA}_{\text{$\sigma$-closed}}$ is the assertion that the universe is not a set-forcing extension of an inner model $\sigma$-closed forcing notion.

Similarly $\text{DDG}_{\text{$\sigma$-closed}}$ is the statement: for all $\sigma$-closed grounds $W_1$, $W_2$ of $V$ there is a $\sigma$-closed ground $U$ (of $V$) contained in $W_1\cap W_2$. Of course, $\text{GA}\rightarrow \text{GA}_{\text{$\sigma$-closed}}$. How about the converse? That is, are there models of ZFC satisfying $\neg\text{GA}+\text{GA}_{\text{$\sigma$-closed}}$ or $\neg\text{GA}+\text{GA}_{\text{ccc}}$? This question arises from Reitz - The ground axiom and when it was published the question was still open.

Moreover, I'm not able to exhibit models in which

- $\text{DDG}_{\text{$\sigma$-closed}}$ fails,
- $\text{DDG}_{\text{ccc}}$ fails.

The last two questions are probably easier.

Outline of inner model theory) and $\mathrm{GA}_{\sigma\mathrm{-closed}}$ (in fact likewise for strategically $\sigma$-closed (in the sense of the ground)) by section 11 ofFine structure from normal iterabilityand Theorem 4.7 ofOrdinal definability in $L[\mathbb{E}]$. If you want I can try to sketch some more info. $\endgroup$ – Farmer S Jan 9 at 23:304more comments