# An overview of virtual large cardinals

This is a talk at the University of Konstanz, July 21, 2023.**Slides**

Measurable cardinals and stronger large cardinals, such as strong and supercompact cardinals, are characterized by the existence of elementary embeddings $j:V\to M$ with the large cardinal as the critical point. Imposing closure properties on $M$, which make $M$ closer to $V$, results in stronger notions. The subject of generic large cardinals bridges the study of the large cardinal hierarchy and forcing by asking what large cardinal notions can be obtained if the embeddings characterizing a given large cardinal notion exist in a forcing extension. For instance, a cardinal $\kappa$ is said to be *generically measurable* if some forcing extension has an embedding $j:V\to M$ with critical point $\kappa$, but $M$ not necessarily contained in $V$. These generic large cardinals are consistency-wise at least as strong as measurable cardinals, but suprisingly very small cardinal like $\omega_1$ can be generally measurable (or supercompact).

Although the most familiar characterizations of large cardinals are in terms of embeddings on the whole universe $V$, all these notions can be equivalently characterized by the existence of set-sized embeddings $j_\lambda:V_\lambda\to M_\lambda$ (usually for cofinally many $\lambda$). Despite the historic interest in generic large cardinals charactized by the existence of embeddings on all of $V$ in the forcing extension, recent attempts to find large cardinal notions for equiconsistency results have resulted in a shift of focus to generic large cardinals defined by asking for the existence of set-sized embeddings characterizing a given large cardinal notion. Surprisingly these set focused generic large cardinals, although they retain many reflecting properties of their actual counterparts, are typically much weaker in consistency strength, and indeed are usually consistent with $L$.

The most unifying concept among these new generic large cardinals has been the notion of *virtual* large cardinals. These are generic large cardinals defined by taking a characterization of a large cardinal notion in terms of the existence of set embeddings $j_\lambda:V_\lambda\to M_\lambda$ and asking that the target $M_\lambda$ exists in $V$ and has the required the closure properties from the perspective of $V$, while only the embedding exists in the forcing extension. The requirement that the target $M_\lambda$ of the embedding is an element of $V$ ensures that the virtual large cardinals are actual large cardinals, they are completely ineffable and more, but usually bounded by the $\omega$-ErdÅ‘s cardinal. The virtual large cardinal hierarchy is therefore compatible with $L$. In fact, the hierarchy even includes virtual versions of large cardinals that are known to be incompatible with ${\rm ZFC}$ (but still believed to be consistent with ${\rm ZF}$).

In this talk, I will give an overview of both the generic and the virtual large cardinal notions that have been defined recently and explain how they arose out of establishing equiconsistency results. I will discuss in which ways the virtual large cardinals mirror the properties of their original counterparts and in which way they differ from them.