Remarkable cardinal
Remarkable cardinals were introduced by Schinder in [1] to provide precise consistency strength of the statement that $L(\mathbb R)$ cannot be modified by proper forcing. A cardinal $\kappa$ is remarkable if for each regular $\lambda>\kappa$, there exists a countable transitive $M$ and an elementary embedding $e:M\rightarrow H_\lambda$ with $\kappa\in \text{ran}(e)$ and also a countable transitive $N$ and an elementary embedding $\theta:M\to N$ such that:
- the critical point of $\theta$ is $e^{-1}(\kappa)$,
- $\text{Ord}^M$ is a regular cardinal in $N$,
- $M=H^N_{\text{Ord}^M}$,
- $\theta(e^{-1}(\kappa))>\text{Ord}^M$.
Remarkable cardinals could be called virtually supercompact, because the following alternative definition is an exact analogue of the definition of supercompact cardinals by Magidor [Mag71]: A cardinal $κ$ is remarkable iff for every $η > κ$, there is $α < κ$ such that in a set-forcing extension there is an elementary embedding $j : V_α → V_η$ with $j(\mathrm{crit}(j)) = κ$.[2]
Remarkable cardinals and the constructible universe
- Remarkable cardinals are downward absolute to $L$. [1]
- If $0^\sharp$ exists, then every Silver indiscernible is remarkable in $L$. [1]
Relations with other large cardinals
- Strong cardinals are remarkable. [1]
- A $2$-iterable cardinal implies the consistency of a remarkable cardinal. [3]
- Remarkable cardinals imply the consistency of $1$-iterable cardinals. [3]
- Remarkable cardinals are totally indescribable. [1]
- Remarkable cardinals are totally ineffable. [1]
- Virtually extendible cardinals are remarkable limits of remarkable cardinals.[2]
- If $κ$ is virtually measurable, then either $κ$ is remarkable in $L$ or $L_κ \models \text{“there is a proper class of virtually measurables”}$.[4]
- Remarkable cardinals are strategic $ω$-Ramsey limits of $ω$-Ramsey cardinals.[4]
References
- Schindler, Ralf-Dieter. Proper forcing and remarkable cardinals. Bull Symbolic Logic 6(2):176--184, 2000. www DOI MR bibtex
- Gitman, Victoria and Shindler, Ralf. Virtual large cardinals. www bibtex
- Gitman, Victoria and Welch, Philip. Ramsey-like cardinals II. J Symbolic Logic 76(2):541--560, 2011. www arχiv MR bibtex
- Nielsen, Dan Saattrup and Welch, Philip. Games and Ramsey-like cardinals. , 2018. arχiv bibtex