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.

Slides

We isolate a new forcing axiom, ${\rm wPFA}$, which is strictly between ${\rm BPFA}$ and ${\rm PFA}$. ${\rm wPFA}$ is equiconsistent with a remarkable cardinal, it implies the failure of $\square_{\omega_1}$, but it is compatible with $\square_\kappa$ for all $\kappa \geq \omega_2$. This is part of joint work with J. Bagaria and V. Gitman.

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.

It is open whether $\Pi^1_1$ determinacy implies the existence of $0^{\#}$ in 3rd order arithmetic, call it $Z_3$. We compute the large cardinal strength of $Z_3$ plus “there is a real $x$ such that every $x$-admissible is an $L$-cardinal.” This is joint work with Yong Cheng.

This is a continuation of the earlier Introduction to remarkable cardinals lecture. The speaker will continue to discuss the various equivalent characterizations of remarkable cardinals and their relationship to other large cardinal notions.

Ralf Schindler introduced remarkable cardinals because he discovered that they are precisely equiconsistent with the statement that the theory of $L(\mathbb R)$ is absolute for proper forcing. The statement that the theory of $L(\mathbb R)$ is absolute for all set forcing is closely related to whether $L(\mathbb R)\models {\rm AD}$. In contrast, remarkable cardinals sit relatively low in the large cardinal hierarchy; for instance, they are downward absolute to $L$. I will discuss the various equivalent characterizations of remarkable cardinals due to Schindler and show where the remarkable cardinals fit into the large cardinal hierarchy using results due to Schindler, Philip Welch and myself.

An extended abstract can be found here.