Remarkable cardinal

Remarkable cardinals were introduced by Schinder in  to provide precise consistency strength of the statement that $L(\mathbb R)$ cannot be modified by proper forcing.

Definitions

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 $j(\mathrm{crit}(j)) = κ$.

Equivalently (theorem 2.4)

• 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)..

Results

Remarkable cardinals and the constructible universe:

• Remarkable cardinals are downward absolute to $L$. 
• If $0^\sharp$ exists, then every Silver indiscernible is remarkable in $L$. 

Relations with other large cardinals:

• Strong cardinals are remarkable. 
• A $2$-iterable cardinal implies the consistency of a remarkable cardinal: Every $2$-iterable cardinal is a limit of remarkable cardinals. 
• 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. 
• Remarkable cardinals are totally indescribable. 
• Remarkable cardinals are totally ineffable. 
• Virtually extendible cardinals are remarkable limits of remarkable cardinals.
• If $κ$ is virtually measurable, then either $κ$ is remarkable in $L$ or $L_κ \models \text{“there is a proper class of virtually measurables”}$.
• Remarkable cardinals are strategic $ω$-Ramsey limits of $ω$-Ramsey cardinals.
• Remarkable cardinals are $Σ_2$-reflecting.

Relation to various set-theoretic principles:

• Equiconsistency with the weak Proper Forcing Axiom:
• 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 $L$.
• It is relatively consistent that ZFC and the generic Vopěnka scheme holds, yet $Ord$ is not definably Mahlo and not even $∆_2$-Mahlo. In such a model, there can be no $Σ_2$-reflecting cardinals and therefore also no remarkable cardinals:
• If $0^♯$ exists, then there is a class-forcing extension $L[G]$ of the constructible universe in which the generic Vopěnka principle holds (so $gVP(κ, \mathbf{Σ_{n+1}})$ and $gVP(Π_n)$ hold for any $κ$ and $n$), but there are no $Σ_2$-reflecting cardinals and hence no remarkable cardinals (or $n$-remarkable cardinals).

Weakly remarkable cardinals

(this section from )

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 $L$.

$n$-remarkable cardinals

$1$-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.