# Topic Archive: set-theoretic geology

Set theory seminarFriday, November 11, 201610:00 amGC 6417

# $V$ need not be a forcing extension of $\mathrm{HOD}$ or of the mantle

CUNY New York City College of Technology

In 1972 Vopenka showed that $V$ is a union of set-generic extensions of $\mathrm{HOD}$ by establishing that every set in $V\setminus\mathrm{HOD}$ is set generic over $\mathrm{HOD}$. It is natural to consider whether that union can be replaced by a single forcing, possibly a proper class, over $\mathrm{HOD}$. In 2012 Friedman showed that $V$ is a class forcing extension of $\mathrm{HOD}$ by a partial order definable in $V$ – however, this leaves open the question of whether such a partial order can be defined in $\mathrm{HOD}$ itself. In this talk I will show that the qualifier ‘in $V$’ is necessary in Friedman’s theorem, by producing a model which is not class generic over $\mathrm{HOD}$ for any forcing definable in $\mathrm{HOD}$.

In the area of set theory known as set-theoretic geology, the mantle $M$ (the intersection of all grounds) is an inner model that enjoys a relationship to $V$ similar to that of $\mathrm{HOD}$, but ‘in the opposite direction’ – every set not in $M$ is omitted by a ground of $V$. Does it follow that we can build $V$ up over $M$ by iteratively adding those sets back in via forcing? In particular, does it follow that $V$ is a class forcing extension of $M$? The example produced in this talk will show that the answer is no – there is a model of set theory $V$ which is not a class forcing extension of $M$ by any forcing definable in $M$.

Set theory seminarFriday, September 9, 201610:00 amGC 6417

# Set-theoretic geology and the downward-directed grounds hypothesis: part II

The City University of New York

I will continue presenting Toshimichi Usuba’s recent proof of the strong downward-directed grounds hypothesis. See the main abstract at Set-theoretic geology and the downward directed ground hypothesis.

Set theory seminarFriday, September 2, 201610:00 amGC 6417

# Set-theoretic geology and the downward-directed grounds hypothesis

The City University of New York

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 $G\subset\mathbb Q\in W$. 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, 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.

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. 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.

Set Theory DayFriday, March 11, 201610:15 amGC 4102 (Science Center)

# Set-theoretic geology: Excavating a local neighborhood of the multiverse

CUNY New York City College of Technology

This talk will give a brief overview of set-theoretic geology, the study of the collection of grounds of $V$. Forcing is naturally viewed as a method for passing from a model $V$ of set theory (the ground model) to an outer model $V[G]$ (the forcing extension). A change in perspective, however, allows us to use forcing to look inward: from a model $V$, we define an inner model $W$ of $V$ to be a ground of $V$ if $W$ is a transitive proper class satisfying ZFC and $V$ can be obtained by forcing over $W$, that is, if $V = W[G]$ for a suitable $W$-generic $G$. For a given model $V$, the collection of all of its ground models forms the context for what we call set-theoretic geology. This second-order collection, consisting of (possibly many) proper classes $W$, nonetheless admits a first-order definition – within a single universe, we have first-order access to an interesting local neighborhood of the set-theoretic multiverse. We will explore this neighborhood, pointing out various geological phenomena including bedrock models, the mantle and the outer core.  This is joint work with Joel David Hamkins and Gunter Fuchs.

Slides

THE MULTIVERSE VIEW – Star Wars Intro

Set theory seminarFriday, February 5, 201610:00 amGC 6417

# Varsovian models

Rutgers University

We compute the mantle of the minimal mouse with a Woodin and strong cardinal. It turns out to be the core model of the mouse. This is a joint work with Ralf Schindler.

CUNY Logic WorkshopFriday, September 7, 201212:00 amGC 6417

# Recent progress on the modal logic of forcing and grounds

The City University of New York

The modal logic of forcing arises when one considers a model of set theory in the context of all its forcing extensions, with “true in all forcing extensions” and“true in some forcing extension” as the accompanying modal operators. In this modal language one may easily express sweeping general forcing principles, such as the assertion that every possibly necessary statement is necessarily possible, which is valid for forcing, or the assertion that every possibly necessary statement is true, which is the maximality principle, a forcing axiom independent of but equiconsistent with ZFC. Similarly, the dual modal logic of grounds concerns the modalities “true in all ground models” and “true in some ground model”. In this talk, I shall survey the recent progress on the modal logic of forcing and the modal logic of grounds. This is joint work with Benedikt Loewe and George Leibman.

The City University of New York
Professor Hamkins (Ph.D. 1994 UC Berkeley) conducts research in mathematical and philosophical logic, particularly set theory, with a focus on the mathematics and philosophy of the infinite.  He has been particularly interested in the interaction of forcing and large cardinals, two central themes of contemporary set-theoretic research.  He has worked in the theory of infinitary computability, introducing (with A. Lewis and J. Kidder) the theory of infinite time Turing machines, as well as in the theory of infinitary utilitarianism and, more recently, infinite chess.  His work on the automorphism tower problem lies at the intersection of group theory and set theory.  Recently, he has been preoccupied with various mathematical and philosophical issues surrounding the set-theoretic multiverse, engaging with the emerging debate on pluralism in the philosophy of set theory, as well as the mathematical questions to which they lead, such as in his work on the modal logic of forcing and set-theoretic geology.