This will be a talk for the CUNY Set Theory Seminar, September 2 and 9, 2016.

In two talks, I shall give a complete detailed account of Toshimichi Usuba’s recent proof of the strong downward-directed grounds hypothesis. This breakthrough result answers what had been for ten years the central open question in the area of set-theoretic geology and leads immediately to numerous consequences that settle many other open questions in the area, as well as to a sharpening of some of the central concepts of set-theoretic geology, such as the fact that the mantle coincides with the generic mantle and is a model of ZFC.

Although forcing is often viewed as a method of constructing larger models extending a given model of set theory, the topic of set-theoretic geology inverts this perspective by investigating how the current set-theoretic universe $V$ might itself have arisen as a forcing extension of an inner model. Thus, an inner model $W\subset V$ is a *ground* of $V$ if we can realize $V=W[G]$ as a forcing extension of $W$ by some $W$-generic filter $G\subset\mathbb{Q}\in W$. It is a consequence of the ground-model definability theorem that every such $W$ is definable from parameters, and from this it follows that many second-order-seeming questions about the structure of grounds turn out to be first-order expressible in the language of set theory.

For example, Reitz had inquired in his dissertation whether any two grounds of $V$ must have a common deeper ground. Fuchs, myself and Reitz introduced the downward-directed grounds hypothesis DDG and the strong DDG, which asserts a positive answer, even for any set-indexed collection of grounds, and we showed that this axiom has many interesting consequences for set-theoretic geology.

Last year, Usuba proved the strong DDG, and I shall give a complete account of the proof, with some simplifications I had noticed. I shall also present Usuba’s related result that if there is a hyper-huge cardinal, then there is a bedrock model, a smallest ground. I find this to be a surprising and incredible result, as it shows that large cardinal existence axioms have consequences on the structure of grounds for the universe.

Among the consequences of Usuba’s result I shall prove are:

- Bedrock models are unique when they exist.
- The mantle is absolute by forcing.
- The mantle is a model of ZFC.
- The mantle is the same as the generic mantle.
- The mantle is the largest forcing-invariant class, and equal to the intersection of the generic multiverse.
- The inclusion relation agrees with the ground-of relation in the generic multiverse. That is, if $N\subset M$ are in the same generic multiverse, then $N$ is a ground of $M$.
- If ZFC is consistent, then the ZFC-provably valid downward principles of forcing are exactly S4.2.
- (Usuba) If there is a hyper-huge cardinal, then there is a bedrock for the universe.

Related topics in set-theoretic geology:

- G. Fuchs, J. D. Hamkins, and J. Reitz, Set-theoretic geology, Annals of Pure and Applied Logic, vol. 166, iss. 4, pp. 464-501, 2015.
- My talk 2016 Aberdeen talk on Set-theoretic pluralism (with slides)
- My other posts on set-theoretic geology
- My London 2011 tutorial on set-theoretic geology

Can you give a super-sketchy basic intuition for the “generic mantle”?

Sure! Suppose that $V$ is the entire set-theoretic universe, satisfying at least ZFC set theory. A class $W\subset V$ is a

groundof $V$, if $V$ is a forcing extension of $W$. Themantleis the intersection of all such grounds. This is not itself necessarily a ground model. For example, in his dissertation, Reitz constructedbottomlessmodels of set theory, which have no minimal grounds. At first, in the geology project we were not able to prove what we had wanted to about the mantle, and so we introduced thegeneric mantle, which seemed to be a more robust class, about which we could prove more. The generic mantle is the intersection of the grounds of all forcing extensions of $V$. So the issue is that if you build a forcing extension $V[G]$ of the universe, then perhaps it has a new ground $W\subset V[G]$, for which $V[G]=W[H]$, and perhaps this new ground crosses the old grounds in such a way that sets in $V$ are not in $W$, although they might have been in every ground of $V$. We had proved many things about the generic mantle, such as that it is the provably largest forcing-invariant class; it is the interestion of the generic multiverse of $V$; it is a model of ZF; and if the strong DDG holds, it is a model of ZFC, and so on. Usuba’s result shows that in fact the strong DDG always holds. It follows that the mantle = generic mantle, and the mantle is a model of ZFC, and so all the other stuff we had wanted to prove about the mantle is now actually proved! The basic situation now is that with Usuba’s result, we don’t need the generic mantle concept any more, since it is equal to the mantle, and all the properties of the generic mantle can simply be stated as theorems about the mantle. Only better, since now we now that the mantle is always a model of ZFC, and is the largest forcing-invariant class, and is the intersection of the generic multiverse, and definable in each model and so on.Thanks, that’s pretty clear! I had another question, but I just answered it by thinking about what you said.

What can one say about this bedrock? (I am hoping that its being “absolute” helps one make real statements about it.)

Yes, this is a really exciting aspect of Usuba’s result, that the existence of large cardinals can imply information about the basic structure of the set-theoretic universe with respect to forcing. This result therefore unifies two major threads in set theory. Basically, if there is a hyper-huge cardinal, then the universe has a smallest ground model, of which it is a forcing extension. The bedrock model will be the same as the mantle (the intersection of grounds) and so it is highly canonical, since it will be forcing-invariant and the largest forcing-invariant class and so on. In general, one cannot prove much about the mantle of an arbitrary model of set theory, since Fuchs, Reitz and I proved that every model of ZFC is the mantle of another model of ZFC. But Usuba’s result places limitations on that if there are these enormous large cardinals—this will be a topic of current investigation. It is conceivable that when there are these sufficiently large cardinals, the mantle (which will be the bedrock) may have some extremely natural properties. Proving or refuting that is definitely the next step here.

Dear Joel,

Is there a reference for this result yet? I’d like to cite it in something!

Also: Has there been any thought to the case where we consider the structure of grounds in a fully second-order setting where we don’t require first-order definability of the ground model in the extension (so going way beyond pseudo-grounds)?

All Best,

Neil

Dear Neil,

As far as I know, the paper has not yet appeared.

Your extended version of the question is interesting. I don’t think we can expect to have directedness in that generality, but I’ll give it some thought. I don’t think we get it even for class forcing grounds.

That would be very interesting (as would other kinds of extension such as when we move from L to L[0#], but with different models and non-forcing extensions).

I take it Usuba’s result applies just as well to pseudo-grounds? From your Aberdeen slides it seems that you just need the relevant cover and approximation properties, and that’s precisely what pseudo-grounds have. I might be missing something here though.