# Restricted notions of set-theoretic geology

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.

• I wonder why this question is not posted on Mathoverflow. It seems better to cross-post the question on MO if you do not receive the answer for long while, at least I think. – Hanul Jeon Jan 9 at 15:10
• I agree with @Hanul that this question might benefit from being on MathOverflow. We can migrate it if you'd like. – Asaf Karagila Jan 9 at 15:49
• Your question on MSE is migrated to there, so this question is now a duplicate of the previous one. – Hanul Jeon Jan 9 at 20:46
• Regarding one of your questions: Assuming large cardinals, the canonical minimal iterable proper class model $M_1$ with one Woodin cardinal models $\mathrm{GA}_{\sigma\mathrm{-closed}}\wedge\neg\mathrm{GA}$. It models $\neg\mathrm{GA}$ by genericity iterations (see Steel's 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 of Fine structure from normal iterability and Theorem 4.7 of Ordinal definability in $L[\mathbb{E}]$. If you want I can try to sketch some more info. – Farmer S Jan 9 at 23:30
• By the way, of course "$\mathbb{P}$ is $\sigma$-closed'' is absolute between $W$ and $V$, but it needn't be for "$\mathbb{P}$ is ccc'', so (I don't know if it's spelled out in Reitz' paper but) when you define $\mathrm{DDG}_{\mathrm{ccc}}$ and $\mathrm{GA}_{\mathrm{ccc}}$, do you mean that $W\models$"$\mathbb{P}$ is ccc" or $V\models$"$\mathbb{P}$ is ccc"? – Farmer S Jan 9 at 23:41

If you add a Cohen real to $$L$$ (or any model of the Ground Axiom), you get a model of the countably closed Ground Axiom, since all intermediate extensions are grounds for Cohen forcing. On the other hand, if you add a Sacks real to $$L$$, you obtain a model with no ccc grounds since Sacks forcing is not ccc and there are no intermediate extensions. So the Sacks extension is a model of the ccc Ground Axiom but not the Ground Axiom.

A counterexample to the countably closed DDG can be obtained by adding mutual generics $$G$$ and $$H$$ for $$\text{Col}(\omega_1,\mathbb R)$$ to $$L(\mathbb R)$$ when $$L(\mathbb R)$$ is a model of DC but not AC. Then $$L(\mathbb R)[G\times H]$$ has $$L(\mathbb R)[G]$$ and $$L(\mathbb R)[H]$$ as countably closed grounds, but their intersection is $$L(\mathbb R)$$, and moreover any countably closed ground of the two models would contain all the reals, and hence be $$L(\mathbb R)$$. Since AC fails in $$L(\mathbb R)$$, it doesn't count as a ground. (Edit: Actually I am remembering now that this example was pointed out by Gunter Fuchs at the inner model theory meeting in Girona in 2018 after I had gone overboard using $$\mathbb P_\text{max}$$.)

To make Asaf happy, if possible, I also point out that assuming ZFC, the countably distributive DDG is true for ZF grounds, though the example above shows it fails for ZFC grounds. (Edit: I mean the intersection of any two ZF grounds closed under countable sequences contains a ZF ground closed under countable sequences. But this need not be a ground of the ZF grounds we started with.) If $$M_0$$ and $$M_1$$ are ZF grounds, there is a ZFC ground $$N$$ contained in both of them by a result of Usuba. Thus $$N(\text{Ord}^\omega)$$ is a common inner model of $$M_0$$ and $$M_1$$ that is closed under $$\omega$$-sequences. Now to show that $$N(\text{Ord}^\omega)$$ is a ground, it suffices by a theorem of Grigorieff to show $$N(\text{Ord}^\omega) = N(X)$$ for some set $$X$$. Let $$\delta$$ be large enough that $$N$$ has the $$\delta$$-cover property, and we claim $$N(\text{Ord}^\omega) \subseteq N(\delta^\omega)$$. To see this, suppose $$\sigma$$ is a countable set of ordinals. Then there is a set of ordinals $$\tau\in N$$ covering $$\sigma$$ of cardinality less than $$\delta$$. Let $$f : \gamma\to \tau$$ be the increasing enumeration of $$\tau$$. Then $$\gamma < \delta$$, so $$\bar \sigma = f^{-1}[\sigma]\in N(\delta^\omega)$$. Therefore $$\sigma = f[\bar \sigma]\in N(\delta^\omega)$$.

I couldn't figure out a counterexample to ccc DDG, though, but I don't doubt that there is one.

• The example with $L(\Bbb R)$ only highlights that the insistence of ZFC in all the models is wrong. – Asaf Karagila Jan 10 at 17:04
• @AsafKaragila I wouldn't dream of insisting on anything. Let one thousand flowers bloom. – Gabe Goldberg Jan 10 at 17:28
• That's offensive to people with allergies. :-P – Asaf Karagila Jan 10 at 23:04
• Regarding ccc DDG, you can start in a model $M$ of CH and add two mutual generics for the forcing to collapse $\omega_1$; call the generics $G_1$ and $G_2$. The point is that the ground model collapse forcing is countable in $M[G_1]$ and $M[G_2]$, so these are ccc grounds for the common extension $M[G_1\times G_2]$. But any common ground would have to be contained in $M$, so the corresponding forcing would need to collapse $\omega_1$. – Miha Habič Jan 11 at 0:13
• @MihaHabič: Cool, so there are even two grounds for Cohen forcing with no common ccc ground. I guess we still don't have a counterexample to the $(\omega_1)^V$-cc DDG, though. – Gabe Goldberg Jan 11 at 0:55