Remarkable cardinal

• Definitions
• Results
• Weakly remarkable cardinals
• $n$-remarkable cardinals

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.

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 $j(\mathrm{crit}(j)) = κ$.(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).

Results

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 $L$.

Weakly remarkable cardinals

(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 $L$.

$n$-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.(Bagaria et al., 2017)

References

1. Schindler, R.-D. (2000). Proper forcing and remarkable cardinals. Bull. Symbolic Logic, 6(2), 176–184. https://doi.org/10.2307/421205
2. Gitman, V., & Shindler, R. Virtual large cardinals. https://ivv5hpp.uni-muenster.de/u/rds/virtualLargeCardinalsEdited5.pdf
3. 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
4. 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
5. Nielsen, D. S., & Welch, P. (2018). Games and Ramsey-like cardinals.
6. Wilson, T. M. (2018). Weakly remarkable cardinals, Erdős cardinals, and the generic Vopěnka principle.

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