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Remarkable cardinal

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:

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))

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:

Relations with other large cardinals:

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

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):

$n$-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)


  1. Schindler, R.-D. (2000). Proper forcing and remarkable cardinals. Bull. Symbolic Logic, 6(2), 176–184.
  2. Gitman, V., & Shindler, R. Virtual large cardinals.
  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.
  4. Gitman, V., & Welch, P. (2011). Ramsey-like cardinals II. J. Symbolic Logic, 76(2), 541–560.
  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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