The virtual large cardinal hierarchy
V. Gitman, S. Dimopoulos, and D. S. Nielsen, “The virtual large cardinal hierarchy,” To appear in Fundamenta Mathematicae.
The study of generic large cardinals, being cardinals that are critical points of elementary embeddings existing in generic extensions, goes back to the 1970's. At that time, the primary interest was the existence of precipitous and saturated ideals on small cardinals like $\omega_1$ and $\omega_2$. Research in this area later moved to the study of more general generic embeddings, both defined on $V$, but also on rank-initial segments of $V$ - these were investigated by e.g. [1] and [2].
The move to virtual large cardinals happened when [3] introduced the remarkable cardinals, which it turned out later were precisely a virtualization of supercompactness. Various other virtual large cardinals were first investigated in [4]. The key difference between virtual large cardinals and generic versions of large cardinals studied earlier is that in the virtual case we require the embedding to be between sets with the target model being a subset of the ground model. These assumptions imply that virtual large cardinals are actual large cardinals: they are at least ineffable, but small enough to exist in $L$. These large cardinals are special because they allow us to work with embeddings as in the higher reaches of the large cardinal hierarchy while being consistent with $V=L$, which enables equiconsistencies at these "lower levels".
To take a few examples, [3] has shown that the existence of a remarkable cardinal is equiconsistent with the statement that the theory of $L(\mathbb R)$ cannot be changed by proper forcing, which was improved to semi-proper forcing in [5]. [6] has shown that the existence of a virtually Vopěnka cardinal is equiconsistent with the hypothesis $$ {\rm ZF}+ \ulcorner\text{${\bf\Sigma}^1_2$ is the class of all $\omega_1$-Suslin sets}\urcorner + \Theta = \omega_2, $$ and (missing reference) has shown that the existence of a virtually Shelah cardinal is equiconsistent with the hypothesis $$ {\rm ZFC} + \ulcorner\text{every universally Baire set of reals has the perfect set property}\urcorner. $$
Kunen's Inconsistency fails for virtual large cardinals in the sense that a forcing extension can have elementary embeddings $j:V_\alpha^V\to V_\alpha^V$ with $\alpha$ much larger than the supremum of the critical sequence. In the theory of large cardinals, Kunen's Inconsistency is for instance used to prove that requiring that $j(\kappa)>\lambda$ in the definition of $\kappa$ being $\lambda$-strong is superfluous. It turns out that the use of Kunen's Inconsistency in that argument is actually essential because versions of virtual strongness with and without that condition are not equivalent. The same holds for virtual versions of other large cardinals where this condition is used in the embeddings characterization. Each of these virtual large cardinals therefore has two non-equivalent versions, with and without the condition.
In this paper, we continue the study of virtual versions of various large cardinals.
We establish some new relationships between virtual large cardinals that were previously studied. We prove the Gitman-Schindler result, alluded to in [4], that virtualizations of strong and supercompact cardinals are equivalent to remarkability. We show how the existence of virtual supercompact cardinals without the $j(\kappa)>\lambda$ condition is related to the existence of virtually rank-into-rank cardinals.
We study virtual versions of Woodin cardinals and introduce the virtual Vopěnka Principle for finite languages. We provide various equivalent characterizations of virtually Woodin cardinals. It follows, from the equivalences, that ${\rm Ord}$ is virtually Woodin if and only if for every class $A$ there is a virtually $A$-extendible cardinal (as defined in [7]), equivalently, there is a stationary class of virtually $A$-extendible cardinals. Recall from [7] that the virtual Vopěnka Principle holds if and only if for every class $A$ there is a proper class of weakly virtually $A$-extendible cardinals. It follows from arguments in [7] that the virtual Vopěnka Principle for finite languages holds if and only if for every class $A$ there is a weakly virtually $A$-extendible cardinal. We show that ${\rm Ord}$ is virtually Vopěnka for finite languages if and only if ${\rm Ord}$ is faintly pre-Woodin, a weakening of the notion of virtual Woodinness.
We study a virtual version of Berkeley cardinals, a large cardinal known to be inconsistent with ${\rm ZFC}$. We show that if there are no virtually Berkeley cardinals, then ${\rm Ord}$ is virtually Woodin if and only if ${\rm Ord}$ is faintly pre-Woodin if and only if the virtual Vopěnka Principle for finite languages holds. In this situation, the virtual Vopěnka Principle for finite languages is equivalent to the virtual Vopěnka Principle. However, we will use virtual Berkeley cardinals to separate the two principles. It follows also that if the virtual Vopěnka Principle holds and ${\rm Ord}$ is not Mahlo (in particular, ${\rm Ord}$ is not virtually Woodin), then there is a virtually Berkeley cardinal, but as pointed out by the anonymous referee, it is possible to have that the virtual Vopěnka Principle holds and ${\rm Ord}$ is Mahlo, but ${\rm Ord}$ is not virtually Woodin.
References
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