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In my previous question Set-theoretic geology: controlled erosion? and the great answer by Jonas Reitz, I have learned a few things, starting from the awareness that I understand the fine-grain structure of Set Theoretic Geology even less than I had assumed.

That is, of course, good news: more to learn!

The second thing I have learned is:

if I want to understand more, I have to start from the STRUCTURAL STANDPOINT, ie I have to grasp, given a transitive model M (I could do away with that, by starting from V, but I prefer concrete set models), the structure of the partial order of grounds of $M$.

To be more specific, Let us begin with $GROUNDS(M)$, and take a look at its structure: it is a partial order, and looks like that it is directed.

So, given two grounds, say $G_1$ and $G_2$, there is a third G which refines both.

Joel's Modal Logic of Forcing is $S4.2$ (please correct me if I am wrong!), which makes sense to me: this logic corresponds exactly to directed partial pre-orders.

But here is where things become quite hazy to me: what about actual meets?

QUESTIONS

  1. When $GROUNDS(M)$ has the structure of a meet-semilattice?
  2. When is $GROUNDS(M)$ equipped with a full lattice structure?
  3. When $GROUNDS(M)$, assuming 1 and 2, is a complete (sups, infs) lattice?

More related questions:

$GROUNDS(M)$ is a subclass of $TM(M)$, ie the class (set) of transitive sub-models of $M$, so it makes sense to loosen the questions above by asking when the infs and sups asked for are not part of the directed order, but still exist in $TM(M)$.

Any answer to any or some of the questions is welcome.

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  • $\begingroup$ I mean, the intersection of any set-many grounds is a ground, is there anything left to say? $\endgroup$
    – Asaf Karagila ♦
    Commented Sep 13, 2020 at 17:31
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    $\begingroup$ Hmm. I could be misremembering, then. But if $W$ and $W'$ are both grounds, and $W\cap W'$ is a model of ZFC which contains a ground $U$, then it is a model of ZFC between $U$ and $V$, where $V=U[G]$ for some set-generic filter $G$. So by Vopenka's theorem $W\cap W'$ is a ground. $\endgroup$
    – Asaf Karagila ♦
    Commented Sep 13, 2020 at 17:42
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    $\begingroup$ No good deed goes unpunished. I'm sure someone will come soon that can write a much better answer to all of your questions. I, on the other hand, need to clean the kitchen and cook the hell out of some chicken. $\endgroup$
    – Asaf Karagila ♦
    Commented Sep 13, 2020 at 17:47
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    $\begingroup$ @Asaf It's not true that the intersection of two grounds is a ground since it might not satisfy ZFC. You can get an example using the technique from my answer here by changing the word "generic" in the second sentence to "Cohen generic" and working in a big collapse extension: mathoverflow.net/questions/297756/… $\endgroup$
    – Gabe Goldberg
    Commented Sep 14, 2020 at 22:35
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    $\begingroup$ @MircoA.Mannucci Another thing you can do is get a situation where you have two grounds whose intersection is $L(\mathbb R)$, and in this case assuming there is no $\omega_1$-sequence of reals in $L(\mathbb R)$, there cannot be a largest common ground, since any inner model of ZFC contained in $L(\mathbb R)$ has countably many reals, and hence has a forcing extension contained in $L(\mathbb R)$. $\endgroup$
    – Gabe Goldberg
    Commented Sep 14, 2020 at 22:50

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Mirco, this is also a fantastic question - the structure of the grounds as a partial order seems to be a very basic aspect of forcing that is not entirely understood. Once again I don’t have a complete answer, but I can provide some background & a few observations.

Intersection of grounds. As pointed out in the comments, it is not the case that the intersection of grounds is (necessarily) a ground - the intersection may fail to satisfy ZFC. However, the intersection does contain a ground (see Directedness)

Directedness. Theorem (Usuba): The grounds are downward-set-directed (that is, the intersection of any collection of set-many grounds contains a further ground).
This fundamental result resolved a number of open questions in Set Theoretic Geology - not least of which is the lovely fact that the Mantle (the intersection of all grounds) is always a model of ZFC.

Meets. Since we can get below any set-indexed collection of grounds, it’s natural to ask whether there is a unique, largest such ground below such a collection. I believe the answer is no (I think the example linked in comments by @gabe-goldberg Intersection of two generic extensions may be a good candidate for a counterexample, but I haven’t thought it through).

The Ground Axiom. The ground axiom states “there are no grounds except V itself” -- if this is the case, then GROUND(V) is trivial.

Least element. The Mantle (the intersection of all grounds) is, by Usuba’s result, a model of ZFC. If GROUND(V) has a least element, it is equal to the Mantle - this will happen exactly when V is a set forcing extension of a model of the Ground Axiom. If GROUND(V) does not have a least element, it may be the case that moving from the Mantle to V can be accomplished by class forcing. Finally, it may be the case that V is not even a class forcing extension of the Mantle (by any class forcing definable in the Mantle) -- see http://jdh.hamkins.org/the-universe-need-not-be-a-class-forcing-extension-of-hod/ .

With regards to your final “more related questions”, I believe that if infs or sups exist then they must be ground models, by an argument similar to @asaf-karagila ‘s in the comments - we can use directedness to get below a collection of grounds, so if an inf exists it is an intermediate model of ZFC between a ground and an extension, hence it is also a ground.

None of the above resolves your three Questions - under what circumstances do we get a nice structure on GROUND(V)? For example, if we start in a model of the Ground Axiom and carry out some forcing, what is the relationship between properties of the forcing we choose and the structure of grounds in the resulting extension? I really like this line of thinking as an avenue for understanding structural properties of forcing.

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  • $\begingroup$ I think that it's safe to assume that the user known as Asaf Karagila is in fact Asaf Karagila. No need to refer to them as "@asaf-karagila" (which will not notify them, and even if it would, there shouldn't be a hyphen there anyway)... $\endgroup$
    – Asaf Karagila ♦
    Commented Sep 17, 2020 at 7:03
  • $\begingroup$ Noted! Thanks Asaf $\endgroup$
    – jonasreitz
    Commented Sep 17, 2020 at 11:51

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