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Remarkable cardinals were introduced by Schinder in (Schindler, 2000) 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 $j(\mathrm{crit}(j)) = κ$.(Gitman & Shindler, n.d.)

Equivalently (theorem 2.4(Bagaria et al., 2017))

- For every $η > κ$ and every $a ∈ V_η$, there is $α < κ$ such that in $V^{Coll(ω,<κ)}$ there is an elementary embedding $j : V_α → V_η$ with $j(crit(j)) = κ$ and $a ∈ range(j)$.
- For every $η > κ$ in $C^{(1)}$ and every $a ∈ V_η$, there is $α < κ$ also in $C^{(1)}$ such that in $V^{Coll(ω,<κ)}$ there is an elementary embedding $j : V_α → V_η$ with $j(crit(j)) = κ$ and $a ∈ range(j)$.
- There is a proper class of $η > κ$ such that for every $η$ in the class, there is $α < κ$ such that in $V^{Coll(ω,<κ)}$ there is an elementary embedding $j : V_α → V_η$ with $j(crit(j)) = κ$

Note: the existence of any such elementary embedding in $V^{Coll(ω,<κ)}$ is equivalent to the existence of such elementary embedding in any forcing extension (see Elementary_embedding#Absoluteness).(Bagaria et al., 2017).

Remarkable cardinals and the constructible universe:

- Remarkable cardinals are downward absolute to $L$. (Schindler, 2000)
- If $0^\sharp$ exists, then every Silver indiscernible is remarkable in $L$. (Schindler, 2000)

Relations with other large cardinals:

- Strong cardinals are remarkable. (Schindler, 2000)
- A $2$-iterable cardinal implies the consistency of a remarkable cardinal: Every $2$-iterable cardinal is a limit of remarkable cardinals. (Gitman & Welch, 2011)
- Remarkable cardinals imply the consistency of $1$-iterable cardinals: If there is a remarkable cardinal, then there is a countable transitive model of ZFC with a proper class of $1$-iterable cardinals. (Gitman & Welch, 2011)
- Remarkable cardinals are totally indescribable. (Schindler, 2000)
- Remarkable cardinals are totally ineffable. (Schindler, 2000)
- Virtually extendible cardinals are remarkable limits of remarkable cardinals.(Gitman & Shindler, n.d.)
- If $κ$ is virtually measurable, then either $κ$ is remarkable in $L$ or $L_κ \models \text{“there is a proper class of virtually measurables”}$.(Nielsen & Welch, 2018)
- Remarkable cardinals are strategic $ω$-Ramsey limits of $ω$-Ramsey cardinals.(Nielsen & Welch, 2018)
- Remarkable cardinals are $Σ_2$-reflecting.(Wilson, 2018)

Equiconsistency with the weak Proper Forcing Axiom:(Bagaria et al., 2017)

- If there is a remarkable cardinal, then $\text{wPFA}$ holds in a forcing extension by a proper poset.
- If $\text{wPFA}$ holds, then $ω_2^V$ is remarkable in $L$.

(this section from (Wilson, 2018))

A cardinal $κ$ is weakly remarkable iff for every $η > κ$, there is $α$ such that in a set-forcing extension there is an elementary embedding $j : V_α → V_η$ with $j(\mathrm{crit}(j)) = κ$. (the condition $α < κ$ is dropped)

A cardinal is remarkable iff it is weakly remarkable and $Σ_2$-reflecting.

The existence of non-remarkable weakly remarkable cardinals is equiconsistent to the existence of an $ω$-Erdős cardinal (equivalent assuming $V=L$; Baumgartner definition of $ω$-Erdős cardinals):

- Every $ω$-Erdős cardinal is a limit of non-remarkable weakly remarkable cardinals.
- If $κ$ is a non-remarkable weakly remarkable cardinal, then some ordinal greater than $κ$ is an $ω$-Erdős cardinal in $L$.

$1$-remarkability is equivalent to remarkability. A cardinal is
virtually $C^{(n)}$-extendible iff it is $n + 1$-remarkable (virtually
extendible cardinals are virtually $C^{(1)}$-extendible). A cardinal is
called **completely remarkable** iff it is $n$-remarkable for all $n
> 0$. Other definitions and properties in Extendible#Virtually
extendible
cardinals.(Bagaria et al., 2017)

- Schindler, R.-D. (2000). Proper forcing and remarkable cardinals.
*Bull. Symbolic Logic*,*6*(2), 176–184. https://doi.org/10.2307/421205 - Gitman, V., & Shindler, R.
*Virtual large cardinals*. https://ivv5hpp.uni-muenster.de/u/rds/virtualLargeCardinalsEdited5.pdf - Bagaria, J., Gitman, V., & Schindler, R. (2017). Generic Vopěnkaś Principle, remarkable cardinals, and the weak Proper Forcing Axiom.
*Arch. Math. Logic*,*56*(1-2), 1–20. https://doi.org/10.1007/s00153-016-0511-x - Gitman, V., & Welch, P. (2011). Ramsey-like cardinals II.
*J. Symbolic Logic*,*76*(2), 541–560. http://boolesrings.org/victoriagitman/files/2011/08/ramseylikecardinalsii.pdf - Nielsen, D. S., & Welch, P. (2018).
*Games and Ramsey-like cardinals*. - Wilson, T. M. (2018).
*Weakly remarkable cardinals, Erdős cardinals, and the generic Vopěnka principle*.