Difference between revisions of "Remarkable"

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 $j(crit(j)) = κ$

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. 
• Remarkable cardinals imply the consistency 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.

Weakly remarkable cardinals

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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

(this section from )

Definition:

• A cardinal $κ$ is $n$-remarkable, for $n > 0$, iff for every $η > κ$ in $C^{(n)}$ , there is $α<κ$ also in $C^{(n)}$ such that in $V^{Coll(ω, < κ)}$, there is an elementary embedding $j : V_α → V_η$ with $j(\mathrm{crit}(j)) = κ$.
• A cardinal is completely remarkable iff it is $n$-remarkable for all $n > 0$.

$1$-remarkability is equivalent to remarkability.

Results:

• Every $n$-remarkable cardinal is in $C^{(n+1)}$.
• Every $n+1$-remarkable cardinal is a limit of $n$-remarkable cardinals.
• Completely remarkable cardinals can exist in $L$ and the consistency of a completely remarkable cardinal follows from a $2$-iterable cardinal.
• In relation to Generic Vopěnka's Principle:
• $gVP(Π_n) \iff gVP(κ, \mathbf{Σ_{n+1}})$ for some $κ \iff$ There is an $n$-remarkable cardinal
• $gVP(Π n ) \iff gVP(κ, \mathbf{Σ_{n+1}})$ for a proper class of $κ \iff$ There is a proper class of $n$-remarkable cardinals

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