# Blog Archives

These are the items posted in this seminar, currently ordered by their post-date, rather than by the event date. We will create improved views in the future. In the meantime, please click on the Seminar menu item above to find the page associated with this seminar, which does have a more useful view order.# Rigidity properties of precipitous ideals

An ideal $I$ on a set $X$ is called *rigid* if forcing with $\mathcal{P}(X)/I$ produces an extension $V[G]$ in which there is a unique $V$-generic filter for $\mathcal{P}(X)/I$. Note that this implies that $\mathcal{P}(X)/I$ has only the trivial automorphism. As part of his analysis of $\mathbb{P}_{\text{max}}$ forcing, Woodin showed that under $\text{MA}_{\omega_1}$, every normal, uniform, saturated ideal on $\omega_1$ is rigid. Indeed, in all previously known models which have rigid precipitous ideals on $\omega_1$ one also has $\lnot\text{CH}$. This leaves open the question: is it consistent to have $\text{CH}$ along with a rigid precipitous ideal on $\omega_1$? I will discuss some recent work which shows that if $\kappa$ is almost huge then there is a forcing extension $V[G]$ in which $\kappa=\omega_1^{V[G]}$, there is a rigid presaturated ideal $I$ on $\omega_1^{V[G]}$ and $\text{CH}$ holds. The proof of this result involves a certain coding forcing first used in the Friedman-Magidor results on the number of normal measures. This is joint work with Sean Cox and Monroe Eskew.

# Virtual large cardinals

Given a very large cardinal property $\mathcal A$, e.g. supercompact or extendible, characterized by the existence of suitable set-sized embeddings, we define that a cardinal $\kappa$ is *virtually* $\mathcal A$ if the embeddings characterizing $\mathcal A$ exist in some set-forcing extension. In this terminology, the remarkable cardinals introduced by Schindler, which he showed to be equiconsistent with the absoluteness of the theory of $L(\mathbb R)$ under proper forcing, are virtually supercompact. We introduce the notions of *virtually extendible*, *virtually $n$-huge*, and *virtually rank-into-rank* cardinals and study their properties. In the realm of virtual large cardinals, we can even go beyond the Kunen Inconsistency because it is possible that in a set-forcing extension there is an embedding $j:V_\delta^V\to V_\delta^V$ with $\delta>\lambda+1$, where $\lambda$ is the supremum of the critical sequence. The virtual large cardinals are much smaller than their (possibly inconsistent) counterparts. Silver indiscernibles possess all the virtual large cardinal properties we will consider, and indeed the large cardinals are downward absolute to $L$. We give a tight measure on the consistency strength of the virtual large cardinals in terms of the $\alpha$-iterable cardinals hierarchy. Virtual large cardinals can be used, for instance, to measure the consistency strength of the *Generic Vopěnka’s Principle*, introduced by Bagaria, Schindler, and myself, which states that for every proper class $\mathcal C$ of structures of the same type, there are $B\neq A$ both in $\mathcal C$ such that $B$ embeds into $A$ in some set-forcing extension. This is joint work with Ralf Schindler.

# Connections between a forcing class and its modal logic

Every definable forcing class $\Gamma$ gives rise to a corresponding modal logic. Mapping the ordered structure given by a ground model $W$ and its $\Gamma$-forcing extensions $W[G]$ to finite ordered structures (frames) in a class which can be characterized by a particular modal theory $\mathsf{S}$ reveals connections between the modal logic of $\Gamma$-forcing and the modal theory $\mathsf{S}$. For example, in 2008, Hamkins and Loewe showed that if ${\rm ZFC}$ is consistent, then the ${\rm ZFC}$-provably valid principles of the class of all forcing are precisely the assertions of the modal theory $\mathsf{S4.2}$. In this talk I describe how specific statements of set theory are used as `controls’ to achieve this mapping for various classes of forcing, giving new results for modal logics of forcing classes such as Cohen forcing, collapse forcing and ${\rm CH}$-preserving forcing.

# Weak compactness without inaccessibility

Weakly compact cardinals are known to have a multitude of equivalent characterizations. The weakly compact cardinals were introduced by Hanf and Tarski in the 1960s as cardinals for which certain infinitary languages satisfied a form of the compactness theorem. Since then, they have been shown to be equivalently characterized as inaccessible cardinals that have, for example, the tree property, or the Keisler extension property, or the $\Pi^1_1$ indescribability property, or the weakly compact embedding property–meaning that they have elementary embeddings with critical point $\kappa$ defined on various transitive sets of size $\kappa$. In this talk we discuss what happens if one drops the requirement that $\kappa$ is inaccessible from weak compactness. We focus especially on the weakly compact embedding property and investigate its relation to the other properties mentioned above. This is joint work with Brent Cody, Sean Cox, and Joel Hamkins, and our results extend prior work of William Boos.

# Minimal models of second-order set theories

Everyone knows that there is a least transitive model of ZFC. Is the same true for second-order set theories? The main result of this talk is that the answer is no for Kelley-Morse set theory. Another notion of minimality we will consider is being the least model with a fixed first-order part. We will see that no countable model of ZFC has a least KM-realization. Along the way, we will look at the analogous questions for Gödel-Bernays set theory.

# Killing measurable and supercompact cardinals softly

This talk follows the theme of killing-them-softly between set-theoretic universes. The main theorems in this theme show how to force to reduce the large cardinal strength of a cardinal to a specified desired degree, for a variety of large cardinals including inaccessible, Mahlo, measurable and supercompact. The killing-them-softly theme is about both forcing and the gradations in large cardinal strength. This talk will focus on measurable and supercompact cardinals, and follows the larger theme of exploring interactions between large cardinals and forcing which is central to modern set theory.

# A position in infinite chess with game value $\omega^4$

I present a position in infinite chess with game value $\omega^4$. Informally speaking, this means that one side can force a win in $\omega^4$ many moves, or that the game is equivalent to the game of counting down from $\omega^4$. This result, joint with Hamkins and Evans, improved on the previous best result of $\omega^3$.

More information can be found at: http://jdh.hamkins.org/tag/infinite-chess.

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

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.

# Joint Laver diamonds

Say that a collection of Laver functions is jointly Laver if the functions can guess their targets simultaneously using just a single elementary embedding between them. In this talk we shall examine the notion of jointness in the simplest case of measurable cardinals, giving both equiconsistency results for the existence of large jointly Laver families and separating the existence of small such families from large ones. We shall also comment on how these results transfer to larger large cardinals, such as supercompact and strong cardinals, and, perhaps, how the notion of jointness may be interpreted for guessing principles not connected with large cardinals.

# Set Theory Day March 11, 2016 at GC

We announce a one-day conference, *Set Theory Day* on Friday, March 11, 2016, in celebration of the 50th birthday of Prof. Joel David Hamkins. The conference will take place at the CUNY Graduate Center and will last all day, with many of Prof. Hamkins’s former students giving talks on their work. All interested mathematicians are invited to attend. The schedule is posted on nylogic.org/set-theory-day. A PDF program is here.