Remarkable cardinal
Remarkable cardinals were introduced by Schinder in [1] to provide precise consistency strength of the statement that $L(\mathbb R)$ cannot be modified by proper forcing.
Contents
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 $j(\mathrm{crit}(j)) = κ$.[2]
Equivalently (theorem 2.4[3])
- 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).[3].
Results
Remarkable cardinals and the constructible universe:
- Remarkable cardinals are downward absolute to $L$. [1]
- If $0^\sharp$ exists, then every Silver indiscernible is remarkable in $L$. [1]
Relations with other large cardinals:
- Strong cardinals are remarkable. [1]
- A $2$-iterable cardinal implies the consistency of a remarkable cardinal: Every $2$-iterable cardinal is a limit of remarkable cardinals. [4]
- 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. [4]
- Remarkable cardinals are totally indescribable. [1]
- Remarkable cardinals are totally ineffable. [1]
- Virtually extendible cardinals are remarkable limits of remarkable cardinals.[2]
- If $κ$ is virtually measurable, then either $κ$ is remarkable in $L$ or $L_κ \models \text{“there is a proper class of virtually measurables”}$.[5]
- Remarkable cardinals are strategic $ω$-Ramsey limits of $ω$-Ramsey cardinals.[5]
- Remarkable cardinals are $Σ_2$-reflecting.[6]
Relation to various set-theoretic principles:
- Equiconsistency with the weak Proper Forcing Axiom:[3]
- 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:[7]
- 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 [6])
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.[3]
References
- Schindler, Ralf-Dieter. Proper forcing and remarkable cardinals. Bull Symbolic Logic 6(2):176--184, 2000. www DOI MR bibtex
- Gitman, Victoria and Shindler, Ralf. Virtual large cardinals. www bibtex
- Bagaria, Joan and Gitman, Victoria and Schindler, Ralf. Generic {V}opěnka's {P}rinciple, remarkable cardinals, and the weak {P}roper {F}orcing {A}xiom. Arch Math Logic 56(1-2):1--20, 2017. www DOI MR bibtex
- Gitman, Victoria and Welch, Philip. Ramsey-like cardinals II. J Symbolic Logic 76(2):541--560, 2011. www arχiv MR bibtex
- Nielsen, Dan Saattrup and Welch, Philip. Games and Ramsey-like cardinals. , 2018. arχiv bibtex
- Wilson, Trevor M. Weakly remarkable cardinals, Erdős cardinals, and the generic Vopěnka principle. , 2018. arχiv bibtex
- Gitman, Victoria and Hamkins, Joel David. A model of the generic Vopěnka principle in which the ordinals are not Mahlo. , 2018. arχiv bibtex