Remarkable cardinals were introduced by Schinder in
{% cite Schindler2000 %} 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](Supercompact.md "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 $j(\mathrm{crit}(j)) =
κ$.{% cite Gitmana %}
Equivalently (theorem
2.4{% cite Bagaria2017a %})
- 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](Elementary%20embedding.md#Absoluteness "Elementary embedding")).{% cite Bagaria2017a %}.
## Results
Remarkable cardinals and the constructible universe:
- Remarkable cardinals are downward absolute to $L$.
{% cite Schindler2000 %}
- If
<a href="Zero_sharp" class="mw-redirect" title="Zero sharp">$0^\sharp
lt;/a>
exists, then every Silver indiscernible is remarkable in $L$.
{% cite Schindler2000 %}
Relations with other large cardinals:
- [Strong](Strong.md "Strong")
cardinals are remarkable.
{% cite Schindler2000 %}
- A
<a href="Iterable" class="mw-redirect" title="Iterable">$2$-iterable</a>
cardinal implies the consistency of a remarkable cardinal: Every
$2$-iterable cardinal is a limit of remarkable cardinals.
{% cite Gitman2011a %}
- Remarkable cardinals imply the consistency of
<a href="Iterable" class="mw-redirect" title="Iterable">$1$-iterable cardinals</a>:
If there is a remarkable cardinal, then there is a countable
transitive model of ZFC with a proper class of $1$-iterable
cardinals. {% cite Gitman2011a %}
- Remarkable cardinals are
<a href="Totally_indescribable" class="mw-redirect" title="Totally indescribable">totally indescribable</a>.
{% cite Schindler2000 %}
- Remarkable cardinals are [](Ineffable.md).
{% cite Schindler2000 %}
- [](Extendible.md)
cardinals are remarkable limits of remarkable
cardinals.{% cite Gitmana %}
- If $κ$ is virtually
[measurable](Measurable.md "Measurable"),
then either $κ$ is remarkable in $L$ or $L_κ \models \text{“there
is a proper class of virtually
measurables”}$.{% cite Nielsen2018 %}
- Remarkable cardinals are strategic
$ω$-[Ramsey](Ramsey.md "Ramsey")
limits of $ω$-Ramsey
cardinals.{% cite Nielsen2018 %}
- Remarkable cardinals are
$Σ_2$-reflecting.{% cite Wilson2018 %}
Equiconsistency with the [](Forcing.md):{% cite Bagaria2017a %}
- 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 $L$.
## Weakly remarkable cardinals
(this section from
{% cite Wilson2018 %})
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](Erdos.md "Erdos")
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 $L$.
## Completely remarkable $n$-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.md#Virtually_extendible_cardinals).{% cite Bagaria2017a %}