# Set-theoretic geology and the downward directed grounds hypothesis, Bonn, January 2017

This will be a talk for the University of Bonn Logic Seminar, Friday, January 13, 2017, at the Hausdorff Center for Mathematics.

Abstract. Set-theoretic geology is the study of the set-theoretic universe $V$ in the context of all its ground models and those of its forcing extensions. For example, a bedrock of the universe is a minimal ground model of it and the mantle is the intersection of all grounds. In this talk, I shall explain some recent advances, including especially the breakthrough result of Toshimichi Usuba, who proved the strong downward directed grounds hypothesis: for any set-indexed family of grounds, there is a deeper common ground below them all. This settles a large number of formerly open questions in set-theoretic geology, while also leading to new questions. It follows, for example, that the mantle is a model of ZFC and provably the largest forcing-invariant definable class. Strong downward directedness has also led to an unexpected connection between large cardinals and forcing: if there is a hyper-huge cardinal $\kappa$, then the universe indeed has a bedrock and all grounds use only $\kappa$-small forcing.

Slides

# Recent advances in set-theoretic geology, Harvard Logic Colloquium, October 2016

I will speak at the Harvard Logic Colloquium, October 20, 2016, 4-6 pm.

Abstract. Set-theoretic geology is the study of the set-theoretic universe $V$ in the context of all its ground models and those of its forcing extensions. For example, a bedrock of the universe is a minimal ground model of it and the mantle is the intersection of all grounds. In this talk, I shall explain some recent advances, including especially the breakthrough result of Toshimichi Usuba, who proved the strong downward directed grounds hypothesis: for any set-indexed family of grounds, there is a deeper common ground below them all. This settles a large number of formerly open questions in set-theoretic geology, while also leading to new questions. It follows, for example, that the mantle is a model of ZFC and provably the largest forcing-invariant definable class. Strong downward directedness has also led to an unexpected connection between large cardinals and forcing: if there is a hyper-huge cardinal $\kappa$, then the universe indeed has a bedrock and all grounds use only $\kappa$-small forcing.

Slides

# 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.

# Upward closure and amalgamation in the generic multiverse of a countable model of set theory

• J. D. Hamkins, “Upward closure and amalgamation in the generic multiverse of a countable model of set theory,” RIMS Kyôkyûroku, pp. 17-31, 2016.
@ARTICLE{Hamkins2016:UpwardClosureAndAmalgamationInTheGenericMultiverse,
author = {Joel David Hamkins},
title = {Upward closure and amalgamation in the generic multiverse of a countable model of set theory},
journal = {RIMS {Ky\^oky\^uroku}},
year = {2016},
volume = {},
number = {},
pages = {17--31},
month = {},
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eprint = {1511.01074},
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issn = {1880-2818},
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Abstract. I prove several theorems concerning upward closure and amalgamation in the generic multiverse of a countable transitive model of set theory. Every such model $W$ has forcing extensions $W[c]$ and $W[d]$ by adding a Cohen real, which cannot be amalgamated in any further extension, but some nontrivial forcing notions have all their extensions amalgamable. An increasing chain $W[G_0]\subseteq W[G_1]\subseteq\cdots$ has an upper bound $W[H]$ if and only if the forcing had uniformly bounded essential size in $W$. Every chain $W\subseteq W[c_0]\subseteq W[c_1]\subseteq \cdots$ of extensions adding Cohen reals is bounded above by $W[d]$ for some $W$-generic Cohen real $d$.

This article is based upon I talk I gave at the conference on Recent Developments in Axiomatic Set Theory at the Research Institute for Mathematical Sciences (RIMS) at Kyoto University, Japan in September, 2015, and I am extremely grateful to my Japanese hosts, especially Toshimichi Usuba, for supporting my research visit there and also at the CTFM conference at Tokyo Institute of Technology just preceding it. This article includes material adapted from section section 2 of Set-theoretic geology, joint with G. Fuchs, myself and J. Reitz, and also includes a theorem that was proved in a series of conversations I had with Giorgio Venturi at the Young Set Theory Workshop 2011 in Bonn and continuing at the London 2011 summer school on set theory at Birkbeck University London.