# Topic Archive: indestructibility

Set theory seminarFriday, February 27, 201510:00 amGC 6417

# Remarkable Laver functions

The City University of New York

Since Laver defined and used a Laver function to show that supercompact cardinals can be made indestructible by all $\lt\kappa$-directed closed forcing, Laver-like functions have been defined for various large cardinals and used for lifting embeddings in indestructibility arguments. Laver-like functions are also inherently interesting as guessing principles with affinity to $\diamondsuit$. Supposing that a large cardinal $\kappa$ can be characterized by the existence of some kind of embeddings, a Laver-like function $\ell:\kappa\to V_\kappa$ has, roughly speaking, the property that for any set $a$ in the universe, there is an embedding $j$ of the type characterizing the cardinal such that $j(\ell)(\kappa)=a$. Although Laver-like functions can be forced to exist for almost any large cardinal, only a few large cardinals including supercompact, strong, and extendible, have them outright. I will define the notion of a remarkable Laver function for a remarkable cardinal and show that every remarkable cardinal has a remarkable Laver function. Remarkable cardinals were introduced by Ralf Schindler who showed that a remarkable cardinal is precisely equiconsistent with the property that the theory of $L(\mathbb R)$ is absolute for proper forcing. Time permitting, I will show how the existence of remarkable Laver functions is used in demonstrating indestructibility for remarkable cardinals. This is joint work with Yong Cheng. An extended abstract can be found here.

Set theory seminarFriday, May 16, 201410:00 amGC6417

# The Laver Preparation

I will present the result of Laver demonstrating that the existence of a supercompact cardinal implies the existence of a Laver function, and using this to construct a model with a supercompact cardinal k which remains supercompact in any k-directed-closed forcing extension.

Rutgers University
Grigor Sargsyan is a professor of mathematics at Rutgers University. He received his Ph.D. at UC Berkeley, 2009. His research interests are in logic, set theory, and foundations: descriptive set theory, inner model theory, large cardinals, and forcing.
The City University of New York
Distinguished professor Arthur W. Apter received his B.S. and Ph.D. degrees, both in mathematics, from MIT in 1975 and 1978, respectively. He is a mathematical logician, with a specialization in set theory, specifically large cardinals, forcing, and indestructibility, and he maintains a burgeoning interest in inner model theory, as well. Professor Apter has published well over 100 research articles.
Set theory seminarFriday, January 31, 201410:00 amGC 6417

# Large cardinals need not be large in HOD

The City University of New York

I will demonstrate that a large cardinal need not exhibit its large cardinal nature in HOD. I shall begin with the example of a measurable cardinal that is not measurable in HOD, and then afterward describe how to force more extreme examples, such as a model with a supercompact cardinal, which is not weakly compact in HOD. This is very recent joint work with Cheng Yong.

Set theory seminarFriday, February 1, 201312:00 amGC 5383New location

# Superstrong cardinals are never Laver indestructible, and neither are extendible, almost huge and rank-into-rank cardinals

The City University of New York

Although the large cardinal indestructibility phenomenon, initiated with Laver’s seminal 1978 result that any supercompact cardinal $kappa$ can be made indestructible by $ltkappa$-directed closed forcing and continued with the Gitik-Shelah treatment of strong cardinals, is by now nearly pervasive in set theory, nevertheless I shall show that no superstrong cardinal—and hence also no $1$-extendible cardinal, no almost huge cardinal and no rank-into-rank cardinal—can be made indestructible, even by comparatively mild forcing: all such cardinals $kappa$ are destroyed by $Add(kappa,1)$, by $Add(kappa,kappa^+)$, by $Add(kappa^+,1)$ and by many other commonly considered forcing notions.

This is very recent joint work with Konstantinos Tsaprounis and Joan Bagaria.