Concept of bedrock and mantle in the multiverse view in the philosophy of mathematics

Question

To be clear, I am not a mathematics educated student and I can not follow the details of the technicality of the forcing extension, but I feel that I have a good understanding of the big picture (of course, it may be wrong).

Our lecturer in the philosophy of mathematics course proposed two theorems in the multiverse view in the philosophy of mathematics which have some confusing features to me. The theorems and definitions are as follows :

Definition 1 : If $V$ is a forcing extension of $W$, then $W$ is a ground of $V$.

Definition 2 : $W$ is a bedrock of $V$ if it is a ground of $V$ and minimal with respect to the forcing extension relation.

Theorem 1 (Reitz) : It is relatively consistent with ZFC that the universe $V$ has no bedrock model. So, such models are bottomless.

Definition 3 The mantle $M$ is the intersection of all grounds of a model of ZFC.

Theorem 2 (Usuba) The grounds are set-directed and so the mantle is a model of ZFC. Furthermore, assuming the consistency of extendible cardinals, mantle itself is a ground of the universe, actually the smallest one.

My question : suppose that $V$ is a bottomless model of ZFC, so there exist the mantle of $V$ which is the smallest ground for $V$ and I consider it as the bedrock of $V$. But based on the theorem 1, this model has not the bedrock and this seems contradictory to me. Am I misunderstanding something?

Answer 1

The confusion seems to arise from your statement "the mantle of $V$ which is the smallest ground for $V$." But this not quite right.

In a bottomless model of ZFC, the mantle is not a ground. It is the intersection of all the grounds, but it isn't one of them. In a bottomless model, one can just keep going to deeper and deeper ground models, without ever landing at a minimal ground.

One can easily make a bottomless model by forcing over $L$, say, with the Easton product class forcing to add a Cohen subset to every regular cardinal. The model arising from any final segment of this forcing (from $\kappa$ and above) is a ground of $L[G]$, since only the set-sized forcing below $\kappa$ is missing. Every ground model of $L[G]$ will contain some final segment of the overall forcing, and one can always peel off a few more factors. So the mantle of $L[G]$ will be $L$ itself.

In the general case, we proved in the geology paper that every model of ZFC arises as the mantle of another model of ZFC.

• Fuchs, Gunter; Hamkins, Joel David; Reitz, Jonas, Set-theoretic geology, Ann. Pure Appl. Logic 166, No. 4, 464-501 (2015). ZBL1348.03051.

Note that a bottomless model can have no extendible cardinal, because by Usuba's theorem the mantle would be a ground, so it wouldn't be bottomless.