# Set-theoretic geology and the downward-directed grounds hypothesis, CUNY Set Theory seminar, September 2016

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:

1. Bedrock models are unique when they exist.
2. The mantle is absolute by forcing.
3. The mantle is a model of ZFC.
4. The mantle is the same as the generic mantle.
5. The mantle is the largest forcing-invariant class, and equal to the intersection of the generic multiverse.
6. 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 $M$.
7. If ZFC is consistent, then the ZFC-provably valid downward principles of forcing are exactly S4.2.
8. (Usuba) If there is a hyper-huge cardinal, then there is a bedrock for the universe.

Related topics in set-theoretic geology:

# Pluralism-inspired mathematics, including a recent breakthrough in set-theoretic geology, Set-theoretic Pluralism Symposium, Aberdeen, July 2016

Set-theoretic Pluralism, Symposium I, July 12-17, 2016, at the University of Aberdeen.  My talk will be the final talk of the conference.

Abstract. I shall discuss several bits of pluralism-inspired mathematics, including especially an account of Toshimichi Usuba’s recent proof of the strong downward-directed grounds DDG hypothesis, which asserts that the collection of ground models of the set-theoretic universe is downward directed. This breakthrough settles several of what were the main open questions of set-theoretic geology. It implies, for example, that the mantle is a model of ZFC and is identical to the generic mantle and that it is therefore the largest forcing-invariant class. Usuba’s analysis also happens to show that the existence of certain very large cardinals outright implies that there is a smallest ground model of the universe, an unexpected connection between large cardinals and forcing. In addition to these results, I shall present several other instances of pluralism-inspired mathematics, including a few elementary but surprising results that I hope will be entertaining.

# Superstrong and other large cardinals are never Laver indestructible

• J. Bagaria, J. D. Hamkins, K. Tsaprounis, and T. Usuba, “Superstrong and other large cardinals are never Laver indestructible,” to appear in Archive for Mathematical Logic (special issue in honor of Richard Laver).
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author = {Joan Bagaria and Joel David Hamkins and Konstantinos Tsaprounis and Toshimichi Usuba},
title = {Superstrong and other large cardinals are never {Laver} indestructible},
journal = {to appear in Archive for Mathematical Logic (special issue in honor of Richard Laver)},
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abstract = {},
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eprint = {1307.3486},
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url = {http://jdh.hamkins.org/superstrong-never-indestructible/},
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Abstract.  Superstrong cardinals are never Laver indestructible. Similarly, almost huge cardinals, huge cardinals, superhuge cardinals, rank-into-rank cardinals, extendible cardinals, $1$-extendible cardinals, $0$-extendible cardinals, weakly superstrong cardinals, uplifting cardinals, pseudo-uplifting cardinals, superstrongly unfoldable cardinals, $\Sigma_n$-reflecting cardinals, $\Sigma_n$-correct cardinals and $\Sigma_n$-extendible cardinals (all for $n\geq 3$) are never Laver indestructible. In fact, all these large cardinal properties are superdestructible: if $\kappa$ exhibits any of them, with corresponding target $\theta$, then in any forcing extension arising from nontrivial strategically ${\lt}\kappa$-closed forcing $\mathbb{Q}\in V_\theta$, the cardinal $\kappa$ will exhibit none of the large cardinal properties with target $\theta$ or larger.

The large cardinal indestructibility phenomenon, occurring when certain preparatory forcing makes a given large cardinal become necessarily preserved by any subsequent forcing from a large class of forcing notions, is pervasive in the large cardinal hierarchy. The phenomenon arose in Laver’s seminal result that any supercompact cardinal $\kappa$ can be made indestructible by ${\lt}\kappa$-directed closed forcing. It continued with the Gitik-Shelah treatment of strong cardinals; the universal indestructibility of Apter and myself, which produced simultaneous indestructibility for all weakly compact, measurable, strongly compact, supercompact cardinals and others; the lottery preparation, which applies generally to diverse large cardinals; work of Apter, Gitik and Sargsyan on indestructibility and the large-cardinal identity crises; the indestructibility of strongly unfoldable cardinals; the indestructibility of Vopenka’s principle; and diverse other treatments of large cardinal indestructibility. Based on these results, one might be tempted to the general conclusion that all the usual large cardinals can be made indestructible.

In this article, my co-authors and I temper that temptation by proving that certain kinds of large cardinals cannot be made nontrivially indestructible. Superstrong cardinals, we prove, are never Laver indestructible. Consequently, neither are almost huge cardinals, huge cardinals, superhuge cardinals, rank-into-rank cardinals, extendible cardinals and $1$-extendible cardinals, to name a few. Even the $0$-extendible cardinals are never indestructible, and neither are weakly superstrong cardinals, uplifting cardinals, pseudo-uplifting cardinals, strongly uplifting cardinals, superstrongly unfoldable cardinals, $\Sigma_n$-reflecting cardinals, $\Sigma_n$-correct cardinals and $\Sigma_n$-extendible cardinals, when $n\geq 3$. In fact, all these large cardinal properties are superdestructible, in the sense that if $\kappa$ exhibits any of them, with corresponding target $\theta$, then in any forcing extension arising from nontrivial strategically ${\lt}\kappa$-closed forcing $\mathbb{Q}\in V_\theta$, the cardinal $\kappa$ will exhibit none of the large cardinal properties with target $\theta$ or larger. Many quite ordinary forcing notions, which one might otherwise have expected to fall under the scope of an indestructibility result, will definitely ruin all these large cardinal properties. For example, adding a Cohen subset to any cardinal $\kappa$ will definitely prevent it from being superstrong—as well as preventing it from being uplifting, $\Sigma_3$-correct, $\Sigma_3$-extendible and so on with all the large cardinal properties mentioned above—in the forcing extension.

Main Theorem.

1. Superstrong cardinals are never Laver indestructible.
2. Consequently, almost huge, huge, superhuge and rank-into-rank cardinals are never Laver indestructible.
3. Similarly, extendible cardinals, $1$-extendible and even $0$-extendible cardinals are never Laver indestructible.
4. Uplifting cardinals, pseudo-uplifting cardinals, weakly superstrong cardinals, superstrongly unfoldable cardinals and strongly uplifting cardinals are never Laver indestructible.
5. $\Sigma_n$-reflecting and indeed $\Sigma_n$-correct cardinals, for each finite $n\geq 3$, are never Laver indestructible.
6. Indeed—the strongest result here, because it is the weakest notion—$\Sigma_3$-extendible cardinals are never Laver indestructible.

In fact, each of these large cardinal properties is superdestructible. Namely, if $\kappa$ exhibits any of them, with corresponding target $\theta$, then in any forcing extension arising from nontrivial strategically ${\lt}\kappa$-closed forcing $\mathbb{Q}\in V_\theta$, the cardinal $\kappa$ will exhibit none of the mentioned large cardinal properties with target $\theta$ or larger.

The proof makes use of a detailed analysis of the complexity of the definition of the ground model in the forcing extension.  These results are, to my knowledge, the first applications of the ideas of set-theoretic geology not making direct references to set-theoretically geological concerns.

Theorem 10 in the article answers (the main case of) a question I had posed on MathOverflow, namely, Can a model of set theory be realized as a Cohen-subset forcing extension in two different ways, with different grounds and different cardinals?  I had been specifically interested there to know whether a cardinal $\kappa$ necessarily becomes definable after adding a Cohen subset to it, and theorem 10 shows indeed that it does:  after adding a Cohen subset to a cardinal, it becomes $\Sigma_3$-definable in the extension, and this fact can be seen as explaining the main theorem above.