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.

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    $\begingroup$ 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. $\endgroup$ – Hanul Jeon Jan 9 at 15:10
  • $\begingroup$ I agree with @Hanul that this question might benefit from being on MathOverflow. We can migrate it if you'd like. $\endgroup$ – Asaf Karagila Jan 9 at 15:49
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    $\begingroup$ Your question on MSE is migrated to there, so this question is now a duplicate of the previous one. $\endgroup$ – Hanul Jeon Jan 9 at 20:46
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    $\begingroup$ 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. $\endgroup$ – Farmer S Jan 9 at 23:30
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    $\begingroup$ 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"? $\endgroup$ – 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.

  • $\begingroup$ The example with $L(\Bbb R)$ only highlights that the insistence of ZFC in all the models is wrong. $\endgroup$ – Asaf Karagila Jan 10 at 17:04
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    $\begingroup$ @AsafKaragila I wouldn't dream of insisting on anything. Let one thousand flowers bloom. $\endgroup$ – Gabe Goldberg Jan 10 at 17:28
  • $\begingroup$ That's offensive to people with allergies. :-P $\endgroup$ – Asaf Karagila Jan 10 at 23:04
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    $\begingroup$ 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 $\omega_1$. $\endgroup$ – Miha Habič Jan 11 at 0:13
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    $\begingroup$ @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. $\endgroup$ – Gabe Goldberg Jan 11 at 0:55

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