Ramsey cardinal
Ramsey cardinals were introduced by Erdős and Hajnal in [1]. Their consistency strength lies strictly between $0^\sharp$ and measurable cardinals.
There are many Ramsey-like cardinals with strength between weakly compact and measurable cardinals inclusively. [2, 3, 4, 5, 6]
Contents
- 1 Ramsey cardinals
- 2 Completely Romsey cardinals etc.
- 3 Almost Ramsey cardinal
- 4 Strongly Ramsey cardinal
- 5 Super Ramsey cardinal
- 6 $\alpha$-iterable cardinal
- 7 Mahlo–Ramsey cardinals
- 8 Very Ramsey cardinals
- 9 Virtually Ramsey cardinal
- 10 Super weakly Ramsey cardinal
- 11 $α$-Ramsey cardinal etc.
- 12 M-rank
- 13 References
Ramsey cardinals
Definitions
A cardinal $\kappa$ is Ramsey if it has the partition property $\kappa\rightarrow (\kappa)^{\lt\omega}_2$.
For infinite cardinals $\kappa$ and $\lambda$, the partition property $\kappa\to(\lambda)^n_\gamma$ asserts that for every function $F:[\kappa]^n\to\gamma$ there is $H\subseteq\kappa$ with $|H|=\lambda$ such that $F\upharpoonright[H]^n$ is constant. Here $[X]^n$ is the set of all $n$-elements subsets of $X$. The more general partition property $\kappa\to(\lambda)^{\lt\omega}_\gamma$ asserts that for every function $F:[\kappa]^{\lt\omega}\to\gamma$ there is $H\subseteq\kappa$ with $|H|=\lambda$ such that $F\upharpoonright[H]^n$ is constant for every $n$, although the value of $F$ on $[H]^n$ may be different for different $n$. Indeed, if $\kappa$ is Ramsey, then $\kappa\rightarrow (\kappa)^{\lt\omega}_\lambda$ for every $\lambda<\kappa$. Ramsey cardinals were named in honor of Frank Ramsey, whose Ramsey theorem for partition properties of $\omega$ motivated the generalizations of these to uncountable cardinals. A Ramsey cardinal $\kappa$ is exactly the $\kappa$-Erdős cardinal.
Ramsey cardinals have a number of other characterizations. They may be characterized model theoretically through the existence of $\kappa$-sized sets of indiscernibles for models meeting the criteria discussed below, as well as through the existence of $\kappa$-sized models of set theory without power set with iterable ultrapowers.
Indiscernibles: Suppose $\mathcal A=(A,\ldots)$ is a model of a language $\mathcal L$ of size less than $\kappa$ whose universe $A$ contains $\kappa$ as a subset.
If a cardinal $\kappa$ is Ramsey, then every such model $\mathcal A$ has a $\kappa$-sized set of indiscernibles $H\subseteq\kappa$, that is, for every formula $\varphi(\overline x)$ of $\mathcal L$ and every pair of tuples $\overline \alpha$ and $\overline \beta$ of elements of $H$, we have $\mathcal A\models\varphi (\overline \alpha)\leftrightarrow \varphi(\overline \beta)$. [7]
Good sets of indiscernibles: Suppose $A\subseteq\kappa$ and $L_\kappa[A]$ denotes the $\kappa^{\text{th}}$-level of the universe constructible using a predicate for $A$. A set $I\subseteq\kappa$ is a good set of indiscernibles for the model $\langle L_\kappa[A],A\rangle$ if for all $\gamma\in I$,
- $\langle L_\gamma[A\cap \gamma],A\cap \gamma\rangle\prec \langle L_\kappa[A], A\rangle$,
- $I\setminus\gamma$ is a set of indiscernibles for the model $\langle L_\kappa[A], A,\xi\rangle_{\xi\in\gamma}$.
A cardinal $\kappa$ is Ramsey if and only if for every $A\subseteq\kappa$, there is a $\kappa$-sized good set of indiscernibles for the model $\langle L_\kappa[A], A\rangle$. [8]
$M$-ultrafilters: Suppose a transitive $M\models {\rm ZFC}^-$, the theory ${\rm ZFC}$ without the power set axiom (and using collection and separation rather than merely replacement) and $\kappa$ is a cardinal in $M$. We call $U\subseteq P(\kappa)^M$ an $M$-ultrafilter if the model $\langle M,U\rangle\models$“$U$ is a normal ultrafilter on $\kappa$”. In the case when the $M$-ultrafilter is not an element of $M$, the model $\langle M,U\rangle$ of $M$ together with a predicate for $U$ often fails to satisfy much of ${\rm ZFC}$. An $M$-ultrafilter $U$ is said to be weakly amenable (to $M$) if for every $A\in M$ of size $\kappa$ in $M$, the intersection $A\cap U$ is an element of $M$. An $M$-ultrafilter $U$ is countably complete if every countable sequence (possibly external to $M$) of elements of $U$ has a non-empty intersection (even if the intersection is not itself an element of $M$). A weak $\kappa$-model is a transitive set $M\models {\rm ZFC}^- $ of size $\kappa$ and containing $\kappa$ as an element. A modified ultrapower construction using only functions on $\kappa$ that are elements of $M$ can be carried out with an $M$-ultrafilter. If the $M$-ultrafilter happens to be countably complete, then the standard argument shows that the ultrapower is well-founded. If the $M$-ultrafilter is moreover weakly amenable, then a weakly amenable ultrafilter on the image of $\kappa$ in the well-founded ultrapower can be constructed from images of the pieces of $U$ that are in $M$. The ultrapower construction may be iterated in this manner, taking direct limits at limit stages, and in this case the countable completeness of the $M$-ultrafilter ensures that every stage of the iteration produces a well-founded model. [9] (Ch. 19)
A cardinal $\kappa$ is Ramsey if and only if every $A\subseteq\kappa$ is contained in a weak $\kappa$-model $M$ for which there exists a weakly amenable countably complete $M$-ultrafilter on $\kappa$. [8]
Ramsey cardinals and the constructible universe
Ramsey cardinals imply that $0^\sharp$ exists and hence there cannot be Ramsey cardinals in $L$. [9]
Relations with other large cardinals
- Measurable cardinals are Ramsey and stationary limits of Ramsey cardinals. [1]
- Ramsey cardinals are unfoldable (using the $M$-ultrafilters characterization) and stationary limits of unfoldable cardinals (as they are stationary limits of $\omega_1$-iterable cardinals).
- Ramsey cardinals are stationary limits of completely ineffable cardinals, they are weakly ineffable, but the least Ramsey cardinal is not ineffable. Ineffable Ramsey cardinals are limits of Ramsey cardinals, because ineffable cardinals are $Π^1_2$-indescribable and being Ramsey is a $Π^1_2$-statement.[3]
- There are stationarily many completely ineffable, greatly Erdős cardinals below any Ramsey cardinal.[4]
Weaker Ramsey-like:
- The existence of a Ramsey cardinal is stronger than the existence of a proper class of almost Ramsey cardinals.
- The Ramsey cardinals are precisely the Erdős almost Ramsey cardinals and also precisely the weakly compact almost Ramsey cardinals.
- A Ramsey cardinal is $\omega_1$-iterable and a stationary limit of $\omega_1$-iterable cardinals. This is already true of an $\omega_1$-Erdős cardinal. [4]
- A virtually Ramsey cardinal that is weakly compact is already Ramsey. If $κ$ is Ramsey, then there is a forcing extension destroying this, while preserving that $κ$ is virtually Ramsey. It is open whether virtually Ramsey cardinals are weaker than Ramsey cardinals.[10, 11]
Stronger Ramsey-like:
- If $κ$ is $Π_1$-Ramsey, then the set of Ramsey cardinals less then $κ$ is in the $Π_1$-Ramsey filter on $κ$.[2]
- Strongly Ramsey cardinals are Ramsey and stationary limits of Ramsey cardinals.[3]
- Mahlo–Ramsey cardinals are a direct strengthening of Ramseyness.[4]
Ramsey cardinals and forcing
- Ramsey cardinals are preserved by small forcing. [9]
- Ramsey cardinals $\kappa$ are preserved by the canonical forcing of the ${\rm GCH}$, by fast function forcing, and by the forcing to add a slim $\kappa$-Kurepa tree. [11]
- If $\kappa$ is Ramsey, there is a forcing extension in which $\kappa$ remains Ramsey and $2^\kappa\gt\kappa$.[11, 12]
- If the ${\rm GCH}$ holds and $F$ is a class function on the regular cardinals having a closure point at $\kappa$ and satisfying $F(\alpha)\leq F(\beta)$ for $\alpha<\beta$ and $\text{cf}(F(\alpha))>\alpha$, then there is a cofinality preserving forcing extension in which $\kappa$ remains Ramsey and $2^\delta=F(\delta)$ for every regular cardinal $\delta$. [12]
- There is a forcing extension in which $κ$ is the first cardinal at which the $\mathrm{GCH}$ fails. [11]
- If the existence of Ramsey cardinals is consistent with ZFC, then there is a model of ZFC in which $\kappa$ is not Ramsey, but becomes Ramsey in a forcing extension. [11]
Completely Romsey cardinals etc.
(All information in this section are from [2] unless otherwise noted)
Basic definitions
- $\mathcal{P}(x)$ is the powerset (set of all subsets) of $x$. $\mathcal{P}_k(x)$ is the set of all subsets of $x$ with exactly $k$ elements.
- $f:\mathcal{P}_k(λ) \to λ$ is regressive iff for all $A \in \mathcal{P}_k(λ)$, we have $f(A) < \min(A)$.
- $E$ is $f$-homogenous iff $E \subseteq λ$ and for all $B,C \in \mathcal{P}_k(E)$, we have $f(B) = f(C)$.
$Π_α$-Ramsey and completely Ramsey
Suppose that $κ$ is a regular uncountable cardinal and $I \supseteq \mathcal{P}_{<κ}(κ)$ is an ideal on $κ$. For every $X \subseteq $κ, $X \in \mathcal{R}^+(I)$ iff for every regressive function $f:\mathcal{P}_{<ω}(κ) \to κ$, for every club $C \subseteq κ$, there is a $Y \in I^+f$ such that $Y \subseteq X \cap C$ and $Y$ is homogeneous for $f$.
- $\mathcal{R}(I) = \mathcal{P}(κ) - \mathcal{R}^+(I)$
- $\mathcal{R}^*(I) = \{ X \subseteq κ : κ - X \in \mathcal{R}(I) \}$
A regular uncountable cardinal $κ$ is Ramsey iff $κ \not\in \mathcal{R}(\mathcal{P}_{<κ}(κ))$. If it is Ramsey, we call $\mathcal{R}(\mathcal{P}_{<κ}(κ))$ the Ramsey ideal on $κ$, its dual $\mathcal{R}^*(\mathcal{P}_{<κ}(κ))$ the Ramsey filter and every element of $\mathcal{R}^+(\mathcal{P}_{<κ}(κ))$ a Ramsey subset of $κ$.
For a regular uncountable cardinal $κ$, we define
- $I_{-2}^κ = \mathcal{P}_{<κ}(κ)$
- $I_{-1}^κ = NS_κ$ (the set of non-stationary subsets of $κ$)
- for $n < ω$, $I_n^κ = \mathcal{R}(I_{n-2}^κ)$
- for $α \geq ω$, $I_{α+1}^κ = \mathcal{R}(I_α^κ)$
- for limit ordinal $γ$, $I_γ^κ = \bigcup_{β<γ} \mathcal{R}(I_β^κ)$
Regular uncountable cardinal $κ$ is $Π_α$-Ramsey iff $κ \not\in I_α^κ$ and completely Ramsey iff for all $α$, $κ \not\in I_α^κ$.
If $κ$ is $Π_α$-Ramsey, we call $I_α^κ$ the $Π_α$-Ramsey ideal on $κ$, its dual the $Π_α$-Ramsey filter and every subset of $κ$ not in $I_α^κ$ a $Π_α$-Ramsey subset. If $κ$ is completely Ramsey, we call $I_{θ_κ}^κ$ the completely Ramsey ideal on $κ$, its dual the completely Ramsey filter and every subset of $κ$ not in $I_{θ_κ}^κ$ a completely Ramsey subset. ($θ_κ$ is the least $α$ such that $I_α^κ = I_{α+1}^κ$ — it must exist before $(2^κ)^+$ for every regular uncountable $κ$ — even if the ideals are trivial)
$α$-hyper completely Ramsey and super completely Ramsey
A sequence $⟨f_α:α<κ^+⟩$ of elements of $^κκ$ is a canonical sequence on $κ$ if both
- for all $α, β\in κ$, $α < β$ implies $f_α < f_β$.
- and for any other sequence $⟨g_α:α<κ^+⟩$ of elements of $κ^κ$ such that $\forall_{α < β < κ^κ} g_α < g_β$, we have $\forall_{α < κ^+} f_α < g_α$.
Note four facts:
- If $⟨f_α:α<κ^+⟩$ and $⟨g_α:α<κ^+⟩$ both are canonical sequences on $κ$, then for all $α < κ^+$ there is a club $C_α \subseteq κ$ such that $\forall_{γ \in C_α} f_α(γ) = g_α(γ)$. (All pairs of corresponding elements of two sequences of functions are equal on a club.)
- There are canonical sequences on each regular uncountable cardinal.
- If $⟨h_α:α<κ^+⟩$ is a canonical sequence on $κ$, then for all $α < κ^+$ there is a club $C_α \subseteq κ$ such that $\forall_{η \in C_α} h_α(η) < |η|^+$. (Each function in a sequence takes on a club values with cardinality not greater then argument's.)
- For all $β < κ^+$ there is a club $C_β \subseteq κ$ such that for all uncountable regular $λ \in C_β$, the set $\{ γ < λ : f^λ_{f^κ_β(λ)}(γ) = f^κ_β(γ) \}$ contains a club in $λ$, where $\vec {f^λ}$ and $\vec {f^κ}$ are canonical sequences on $λ$ and $κ$ respectively.
For a regular uncountable cardinal $κ$, let $\vec f = ⟨f_α:α<κ^+⟩$ be the canonical sequence on $κ$.
- $κ$ is 0-hyper completely Ramsey iff $κ$ is completely Ramsey.
- For $α<κ^+$, $κ$ is $α+1$-hyper completely Ramsey iff $κ$ is $α$-hyper completely Ramsey and there is a completely Ramsey subset $X$ such that for all $λ \in X$, $λ$ is $f_α(λ)$-hyper completely Ramsey.
- For $γ \leq κ^+$, $κ$ is $γ$-hyper completely Ramsey iff $κ$ is $β$-hyper completely Ramsey for all $β<γ$.
- $κ$ is super completely Ramsey iff $κ$ is $κ^+$-hyper completely Ramsey.
Terminology
(This subsection compares (Sharpe&Welch, 2011) and (Feng, 1990))
$Π_α$-Ramsey cardinals correspond to $α$-Ramsey and $α$-Ramsey$^s$ in [4].[5] (The “$^s$” stands for “stationary”.[4])
$Π_{2 n}$-Ramsey cardinals are Sharpe-Welch $n$-Ramsey and $Π_{2 n + 1}$-Ramsey cardinals are $n$-Ramsey$^s$.
For infinite $α$, $Π_α$-Ramsey, Sharpe-Welch $α$-Ramsey and $α$-Ramsey$^s$ are identical.
Results
Absoluteness:
- All this properties (being Ramsey itself, $Π_α$-Ramsey, completely Ramsey, $α$-hyper completely Ramsey and super completely Ramsey) are downwards absolute to the Dodd-Jensen core model.
Hierarchy:
- There are stationary many $Π_n$-Ramsey cardinals below each $Π_{n+1}$-Ramsey cardinal.
- If $κ$ is $Π_{α+1}$-Ramsey and $α < κ^+$, then the set of $Π_α$-Ramsey cardinals less then $κ$ is in the $Π_{α+1}$-Ramsey filter on $κ$.
Upper limit of consistency strength:
- Any measurable cardinal is super completely Ramsey and a stationary limit of super completely Ramsey cardinals.
Indescribability:
- If $κ$ is $Π_n$-Ramsey, then $κ$ is $Π_{n+1}^1$-indescribable. If $X \subseteq κ$ is a $Π_n$-Ramsey subset, then $X$ is $Π_{n+1}^1$-indescribable.
- For infinite $α$, if $κ$ is $Π_α$-Ramsey, then $κ$ is $Π^1_{2 ·(1+β)+ 1}$-indescribable for each $β < \min \{α, κ^+\}$ (Transfinite $Π^1_α$-indescribable is defined via finite games.).[4]
- If $κ$ is completely Ramsey, then $κ$ is $Π_1^2$-indescribable.[5]
Equivalence:
Relation with other variants of Ramseyness:
- Strongly Ramsey cardinals are limits of completely Ramsey cardinals, but are not necessarily completely Ramsey themselves.[10]
- Every $(ω+1)$-Ramsey cardinal is a completely Ramsey stationary limit of completely Ramsey cardinals.[6]
- Any $\Pi_2$-Ramsey cardinal is $α$-Mahlo–Ramsey for all $α < κ^+$. [4]
Almost Ramsey cardinal
cf. (Vickers&Welch, 2001)
An uncountable cardinal $\kappa$ is almost Ramsey if and only if $\kappa\rightarrow(\alpha)^{<\omega}$ for every $\alpha<\kappa$. Equivalently:
- $\kappa\rightarrow(\alpha)^{<\omega}_\lambda$ for every $\alpha,\lambda<\kappa$
- For every structure $\mathcal{M}$ with language of size $<\kappa$, there is are sets of indiscernibles $I\subseteq\kappa$ for $\mathcal{M}$ of any size $<\kappa$.
- For every $\alpha<\kappa$, $\eta_\alpha$ exists and $\eta_\alpha<\kappa$.
- $\kappa=\text{sup}\{\eta_\alpha:\alpha<\kappa\}$
($\eta_\alpha$ is the $\alpha$-Erdős cardinal.)
Every almost Ramsey cardinal is a $\beth$-fixed point, but the least almost Ramsey cardinal, if it exists, has cofinality $\omega$. In fact, the least almost Ramsey cardinal is not weakly inaccessible, worldly, or correct. However, if the least almost Ramsey cardinal exists, it is larger than the least $\omega_1$-Erdős cardinal. Any regular almost Ramsey cardinal is worldly, and any worldly almost Ramsey cardinal $\kappa$ has $\kappa$ almost Ramsey cardinals below it.
The existence of a worldly almost Ramsey cardinal is stronger than the existence of a proper class of almost Ramsey cardinals. Therefore, the existence of a Ramsey cardinal is stronger than the existence of a proper class of almost Ramsey cardinals. The existence of a proper class of almost Ramsey cardinals is equivalent to the existence of $\eta_\alpha$ for every $\alpha$. The existence of an almost Ramsey cardinal is stronger than the existence of an $\omega_1$-Erdős cardinal.
The existence of an almost Ramsey cardinal is equivalent to the existence of $\eta^n(\omega)$ for every $n<\omega$. On one hand, if a almost Ramsey cardinal $\kappa$ exists, then $\omega<\kappa$. Then, $\eta_\omega$ is less than $\kappa$. Then, $\eta_{\eta_\omega}$ exists and is less than $\kappa$, and so on. On the other hand, if $\eta^n(\omega)$ exists for every $n<\omega$, then $\text{sup}\{\eta^n(\omega):n<\omega\}$ is almost Ramsey, and in fact the least almost Ramsey cardinal. Note that such a set exists by replacement and has a supremum by union.
The Ramsey cardinals are precisely the Erdős almost Ramsey cardinals and also precisely the weakly compact almost Ramsey cardinals.
If $κ$ is a $2$-weakly Erdős cardinal, then $κ$ is almost Ramsey.[4]
Strongly Ramsey cardinal
Strongly Ramsey cardinals were introduced by Gitman in [3] (all information from there unless otherwise noted). They strengthen the $M$-ultrafilters characterization of Ramsey cardinals from weak $\kappa$-models to $\kappa$-models.
A cardinal $\kappa$ is strongly Ramsey if every $A\subseteq\kappa$ is contained in a $\kappa$-model $M$ for which there exists a weakly amenable $M$-ultrafilter on $\kappa$. An $M$-ultrafilter for a $\kappa$-model $M$ is automatically countably complete since $\langle M,U\rangle$ satisfies that it is $\kappa$-complete and it must be correct about this since $M$ is closed under sequences of length less than $\kappa$.
Properties:
- Super Ramsey cardinals are strongly Ramsey limits of strongly Ramsey cardinals.
- Strongly Ramsey cardinals are limits of completely Ramsey cardinals, but are not necessarily completely Ramsey themselves.[10]
- Every strongly Ramsey cardinal is a stationary limit of almost fully Ramseys.[6]
- Strongly Ramsey cardinals are Ramsey and stationary limits of Ramsey cardinals.
- The least strongly Ramsey cardinal is not ineffable.
- Forcing related properties of strongly Ramsey cardinals are the same as those of Ramsey cardinals described above. [11]
- Strong Ramseyness is downward absolute to $K$. [10]
Super Ramsey cardinal
Super Ramsey cardinals were introduced by Gitman in [3] (all information from there unless otherwise noted). They strengthen one definition of strong Ramseyness.
A weak $\kappa$-model $M$ is a $\kappa$-model if additionally $M^{\lt\kappa}\subseteq M$.
A cardinal $\kappa$ is super Ramsey if and only if for every $A\subseteq\kappa$, there is some $\kappa$-model $M$ with $A\subseteq M\prec H_{\kappa^+}$ such that there is some $N$ and some $\kappa$-powerset preserving nontrivial elementary embedding $j:M\prec N$.
The following are some facts about super Ramsey cardinals:
- Measurable cardinals are super Ramsey limits of super Ramsey cardinals.
- Fully Ramsey cardinals are super Ramsey limits of super Ramsey cardinals.[5]
- Super Ramsey cardinals are strongly Ramsey limits of strongly Ramsey cardinals.
- Super Ramseyness is downward absolute to $K$. [10]
- The required $M$ for a super Ramsey embedding is stationarily correct.
$\alpha$-iterable cardinal
The $\alpha$-iterable cardinals for $1\leq\alpha\leq\omega_1$ were introduced by Gitman in [10]. They form a hierarchy of large cardinal notions strengthening weakly compact cardinals, while weakening the $M$-ultrafilter characterization of Ramsey cardinals. Recall that if $\kappa$ is Ramsey, then every $A\subseteq\kappa$ is contained in a weak $\kappa$-model $M$ for which there exists an $M$-ultrafilter, the ultrapower construction with which may be iterated through all the ordinals. Suppose $M$ is a weak $\kappa$-model and $U$ is an $M$-ultrafilter on $\kappa$. Define that:
- $U$ is $0$-good if the ultrapower is well-founded,
- $U$ is 1-good if it is 0-good and weakly amenable,
- for an ordinal $\alpha>1$, $U$ is $\alpha$-good, if it produces at least $\alpha$-many well-founded iterated ultrapowers.
Using a theorem of Gaifman [13], if an $M$-ultrafilter is $\omega_1$-good, then it is already $\alpha$-good for every ordinal $\alpha$.
For $1\leq\alpha\leq\omega_1$, a cardinal $\kappa$ is $\alpha$-iterable if every $A\subseteq\kappa$ is contained in a weak $\kappa$-model $M$ for which there exists an $\alpha$-good $M$-ultrafilter on $\kappa$.
The $\alpha$-iterable cardinals form a hierarchy of strength above weakly compact cardinals and below Ramsey cardinals.
The $1$-iterable cardinals are sometimes called the weakly Ramsey cardinals.
Results
Lower limit:
- $1$-iterable cardinals are weakly ineffable and stationary limits of completely ineffable cardinals. The least $1$-iterable cardinal is not ineffable. [3]
- Super weakly Ramsey cardinals are weakly Ramsey (=$1$-iterable) limits of weakly Ramsey cardinals.
Upper limit:
- A Ramsey cardinal is $\omega_1$-iterable and a stationary limit of $\omega_1$-iterable cardinals. This is already true of an $\omega_1$-Erdős cardinal. [4]
- If $C ∈ V[H]$, a forcing extension by $\mathrm{Coll}(ω, V_κ)$, is a club in $κ$ of generating indiscernibles for $V_κ$ of order-type $κ$ (like in the definition of Silver cardinals), then each $ξ ∈ C$ is $< ω_1$-iterable.[14]
- $ω_1$-iterable cardinals are strongly unfoldable in $L$.[10]
Hierarchy:
- An $\alpha$-iterable cardinal is $\beta$-iterable and a stationary limit of $\beta$-iterable cardinals for every $\beta<\alpha$. [10]
- For $β > 0$, every $(α, β)$-Ramsey is a $β$-iterable stationary limit of $β$-iterables.
- It is consistent from an $\omega$-Erdős cardinal that for every $n\in\omega$, there is a proper class of $n$-iterable cardinals.
- For an additively indecomposable ordinal $λ ≤ ω_1$, $η_λ$ (the least $λ$-Erdős cardinal) is a limit of $λ$-iterable cardinals and if there is a $λ + 1$-iterable cardinal, then there is a $λ$-Erdős cardinal below it.[14]
- A virtually $n$-huge* cardinal is an $n+1$-iterable limit of $n+1$-iterable cardinals. If $κ$ is $n+2$-iterable, then $V_κ$ is a model of proper class many virtually $n$-huge* cardinals.[14]
- Every virtually rank-into-rank cardinal is an $ω$-iterable limit of $ω$-iterable cardinals.[14]
Between $1$- and $2$-iterable:
- A $2$-iterable cardinal implies the consistency of a remarkable cardinal: Every $2$-iterable cardinal is a limit of remarkable cardinals. [10]
- 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. [10]
- If $κ$ is $2$-iterable, then $V_κ$ is a model of proper class many virtually $C^{(n)}$-extendible cardinals for every $n < ω$, of proper class many virtually Shelah for supercompactness cardinals[14] and of proper class many completely remarkable cardinals.[15]
- Virtually extendible cardinals are 1-iterable limits of 1-iterable cardinals.[14]
Absoluteness:
- $\omega_1$-iterable cardinals imply that $0^\sharp$ exists and hence there cannot be $\omega_1$-iterable cardinals in $L$. For $L$-countable $\alpha$, the $\alpha$-iterable cardinals are downward absolute to $L$. In fact, if $0^\sharp$ exists, then every Silver indiscernible is $\alpha$-iterable in $L$ for every $L$-countable $\alpha$. [10]
- $\alpha$-iterable cardinals $\kappa$ are preserved by small forcing, by the canonical forcing of the ${\rm GCH}$, by fast function forcing, and by the forcing to add a slim $\kappa$-Kurepa tree. If $\kappa$ is $\alpha$-iterable, there is a forcing extension in which $\kappa$ remains $\alpha$-iterable and $2^\kappa\gt\kappa$. [11]
Mahlo–Ramsey cardinals
The property of being Mahlo–Ramsey (MR) is a slight strengthening of Ramseyness introduced in analogy to Mahlo cardinals in [4] (all information from there).
For a regular cardinal $κ$ and a sequence of canonical functions $⟨ f_α : α < κ^+ ⟩$
- $κ$ is $0$-MR iff it is Ramsey.
- $κ$ is $(α + 1 )$-MR iff for any $g : \mathcal{P}_{<ω}(κ) → 2$ there is an $X ∈ NS^+_κ$ such that $X$ is homogeneous for $g$ and $∀_{μ ∈ X} \text{$μ$ is $f_α (μ)$-MR}$.
- $κ$ is $δ$-MR for limit $δ < κ^+$ iff it is $α$-MR for all $α < δ$.
Any $\Pi_2$-Ramsey cardinal is $α$-MR for all $α < κ^+$.
Very Ramsey cardinals
For $X ⊆ κ$ and ordinal $α$, $G_R(X, α)$ is a certain game for two players with finitely many moves defined in (Sharpe&Welch11). $X$ is Sharpe-Welch $\alpha$-Ramsey iff (II) wins $G_R(X, α)$. $G_r(X, α)$ (also defined there) is a modification of the game allowing $1+α$ moves. $X$ is $\alpha$-very Ramsey iff (II) has a winning strategy in $G_r(X, α)$.[4]
For $n < ω$, the games $G_R(X, n)$ and $G_r(X, n)$ coincide.[4]
In analogy to coherent $<α$-very Ramsey, one can define coherent $<α$-very Ramsey cardinals. $α$-very Ramsey cardinals are equivalent to coherent $<α$-very Ramsey cardinals for limit $α$ and to $<(α+1)$-very Ramsey cardinals in general. (This just allows to “subtract one” for successor ordinals.)[6]
Results:
- A cardinal is completely Ramsey iff it is $ω$-very Ramsey.[4, 6]
- If $κ$ is a measurable cardinal, then $κ$ is $κ$-very Ramsey.[4]
- If a cardinal is $ω_1$-very Ramsey (=strategic $ω_1$-Ramsey cardinal), then it is measurable in the core model unless $0^\P$ exists and an inner model with a Woodin cardinal exists.[4, 6]
Additional results from [6]:
- For limit ordinal $α$, every coherent $<ωα$-Ramsey is $ωα$-very Ramsey.
- For any ordinal $α$, every coherent $<α$-very Ramsey is coherent $<α$-Ramsey.
- For limit ordinal $α$, $κ$ is $ωα$-very Ramsey iff it is coherent $<ωα$-Ramsey.
- $κ$ is $λ$-very Ramsey iff it is strategic $λ$-Ramsey for any $λ$ with uncountable cofinality.
Virtually Ramsey cardinal
Virtually Ramsey cardinals were introduced by Sharpe and Welch in [4]. They weaken the good indiscernibles characterization of Ramsey cardinals and were motivated by finding an upper bound on the consistency strength of a variant of Chang's Conjecture studied in [4]. For $A\subseteq\kappa$, define that $\mathscr I=\{\alpha<\kappa\mid$ there is an unbounded good set of indiscernibles $I_\alpha\subseteq\alpha$ for $\langle L_\kappa[A],A\rangle\}$. A cardinal $\kappa$ is virtually Ramsey if for every $A\subseteq\kappa$, the set $\mathscr I$ contains a club of $\kappa$.
Virtually Ramsey cardinals are Mahlo and a virtually Ramsey cardinal that is weakly compact is already Ramsey. If $κ$ is Ramsey, then there is a forcing extension destroying this, while preserving that $κ$ is virtually Ramsey. It is open whether virtually Ramsey cardinals are weaker than Ramsey cardinals.[10, 11]
If κ is virtually Ramsey then κ is greatly Erdős.[4]
Super weakly Ramsey cardinal
(All from [5])
A cardinal $κ$ is super weakly Ramsey iff every $A ⊆ κ$ is contained, as an element, in a weak κ-model $M ≺ H(κ^+)$ for which there exists a $κ$-powerset preserving elementary embedding $j∶ M → N$.
Strength:
- Super weakly Ramsey cardinals are weakly Ramsey (=$1$-iterable) limits of weakly Ramsey cardinals.
- Super weakly Ramsey cardinals are ineffable.
- $ω$-Ramsey cardinals are super weakly Ramsey limits of super weakly Ramsey cardinals.
$α$-Ramsey cardinal etc.
$α$-Ramsey cardinal for cardinal $α$
(All from [5])
For regular cardinal $α ≤ κ$, $κ$ is $α$-Ramsey iff for arbitrarily large regular cardinals $θ$, every $A ⊆ κ$ is contained, as an element, in some weak $κ$-model $M ≺ H(θ)$ which is closed under $<α$-sequences, and for which there exists a $κ$-powerset preserving elementary embedding $j∶ M → N$.
Note that, in the case $α = κ$, a weak $κ$-model closed under $<κ$-sequences is exactly a $κ$-model.
Alternate characterisation:
- For regular uncountable cardinal $α ≤ κ$, $κ$ is $α$-Ramsey iff $κ = κ^{<κ}$ has the $α$-filter property.
- $κ$ is $ω$-Ramsey iff $κ = κ^{<κ}$ has the well-founded $ω$-filter property.
This characterisation works better for singular alpha $α$ (the original one would imply being $α^+$-Ramsey; well-founded $α$-filter property is better for countable cofinality).
Games for definitions
(from [6] unless otherwise noted)
For a weak $κ$-model $\mathcal{M}$, $μ$ is an $\mathcal{M}$-measure iff $(\mathcal{M}, \in, μ) \models \text{“$μ$ is a $κ$-complete ultrafilter on $κ$”}$.
Games $G_1$ and $G_2$ are equivalent when each of two players has a winning strategy in $G_1$ if and only if he has one in $G_2$.
The $α$-Ramsey cardinals are based upon well-founded filter games[5] $wfG^θ_γ(κ)$ (full definition in sources).
- Player I (challenger[5]) gives $\subseteq$-increasing $κ$-models $\mathcal{M}_α ≺ H_θ$,
- player II (judge[5]) gives $\subseteq$-increasing filters $μ_α$ that are $\mathcal{M}_α$-measures
- and II wins iff after $γ$ rounds $μ$ is an $\mathcal{M}$-normal good $\mathcal{M}$-measure for $μ = \bigcup_{α<γ} μ_α$ and $\mathcal{M} = \bigcup_{α<γ} \mathcal{M}_α$.
The games $wfG^{θ_0}_γ(κ)$ and $wfG^{θ_1}_γ(κ)$ are equivalent for any $γ$ with $\mathrm{cof}(γ) \neq ω$ and any regular $θ_0, θ_1 < κ$.
$\mathcal{G}^θ_γ(κ, ζ)$ is a similar family of games (again full definition in sources).
- Each of them lasts up to $γ+1$ rounds
- and player II wins when he does not have to end the game before $γ+1$ rounds pass
- (I gives $\subseteq$-increasing weak $κ$-models
- and II must give normal $\mathcal{M}_α$-measures with additional requirements for limit $α$ (eg. $μ_α$ is $ζ$-good) and for the last move).
For convenience
- $\mathcal{G}^θ_γ(κ) := \mathcal{G}^θ_γ(κ, 0)$
- $\mathcal{G}_γ(κ) := \mathcal{G}^θ_γ(κ)$ whenever $\mathrm{cof}(γ) \neq ω$ as again the existence of winning strategies in these games does not depend upon a specific $θ$.
$\mathcal{G}^θ_γ(κ)$, $\mathcal{G}^θ_γ(κ, 1)$ and $wfG^θ_γ(κ)$ are all equivalent for all limit ordinals $γ \leq κ$. $\mathcal{G}^θ_γ(κ, ζ)$ is equivalent to $\mathcal{G}^θ_γ(κ)$ whenever $\mathrm{cof}(γ) > ω$.
Generalisations
(from [6])
Now we can define $γ$-Ramsey cardinals for any ordinal $γ$ and other variants: Let $κ$ be a cardinal and $γ \leq κ$ an ordinal:
- $κ$ is $γ$-Ramsey iff player I does not have a winning strategy in $\mathcal{G}^θ_γ(κ)$ for all regular $θ > κ$.
- $κ$ is strategic $γ$-Ramsey iff player II does have a winning strategy in $\mathcal{G}^θ_γ(κ)$ for all regular $θ > κ$.
- (Strategic) genuine $γ$-Ramseys and (strategic) normal $γ$-Ramseys are defined analogously, with the additional requirement for the last measure $μ_γ$ to be genuine and normal, respectively.
- $κ$ is $<γ$-Ramsey iff it is $α$-Ramsey for every $α < γ$.
- $κ$ is almost fully Ramsey iff it is $<κ$-Ramsey.
- $κ$ is fully Ramsey iff it is $κ$-Ramsey.
- $κ$ is coherent $<γ$-Ramsey iff it is strategic $<γ$-Ramsey and a single strategy works for player II in $\mathcal{G}_α(κ)$ for every $α < γ$.
- I.e., in a choice of strategies for each $α$, strategies for greater $α$ contain strategies for lesser $α$. Full definition in [6].
(Some of the notions defined in [6] were not new, but gained more convenient names.)
Filter property
(from [5])
$κ$ has the filter property iff for every subset $X$ of $\mathcal{P}(κ)$ of size $≤κ$, there is a $<κ$-complete filter $F$ on $κ$ which measures $X$. For normal filter we talk about normal filter property.
Strengthenings:
- $κ$ has the $γ$-filter property iff player I does not have a winning strategy in $G^θ_γ(κ)$.
- $κ$ has the strategic $γ$-filter property iff player II does have a winning strategy in $G^θ_γ(κ)$.
- $κ$ has the well-founded $(γ, θ)$-filter property iff player I does not have a winning strategy in $wfG^θ_γ(κ)$.
- $κ$ has the well-founded $γ$-filter property iff it has the well-founded $(γ, θ)$-filter property for all regular $θ > κ$.
For $γ_1 > γ_0$, the $γ_1$-filter property implies the $γ_0$-filter property.
Results
Results in the finite case (for $n < ω$):[6]
- For a cardinal $κ=κ^{<κ}$
- $κ$ is weakly compact iff it is 0-Ramsey;
- $κ$ is weakly ineffable iff it is genuine 0-Ramsey;
- $κ$ is ineffable iff it is normal 0-Ramsey. (An uncountable cardinal κ has the normal filter property iff it is ineffable.[5])
- Every $n$-Ramsey $κ$ is $Π^1_{2 n+1}$-indescribable. This is optimal, as $n$-Ramseyness can be described by a $Π^1_{2 n+2}$-formula.
- Every $<ω$-Ramsey cardinal is $∆^2_0$-indescribable.
- Every normal $n$-Ramsey $κ$ is $Π^1_{2 n+2}$-indescribable. This is optimal, as normal $n$-Ramseyness can be described by a $Π^1_{2 n+3}$-formula.
- Every $n+1$-Ramsey is a normal $n$-Ramsey stationary limit of normal $n$-Ramseys and every normal $n$-Ramsey is an $n$-Ramsey stationary limit of $n$-Ramseys.
- Genuine- and normal $n$-Ramseys are downwards absolute to $L$.
- Every $(n+1)$-Ramsey is normal $n$-Ramsey in $L$. Therefore, $<ω$-Ramseys are downwards absolute to $L$.
Results for $ω$-Ramsey:[5]
- $ω$-Ramsey cardinals are super weakly Ramsey limits of super weakly Ramsey cardinals.
- $ω$-Ramsey cardinals are limits of cardinals with the $ω$-filter property (=completely ineffable[6]).
- $ω$-Ramsey cardinals are downwards absolute to $L$. If $0^♯$ exists, then all Silver indiscernibles are $ω$-Ramsey in $L$.
Results for strategic $ω$-Ramsey:[6]
- Virtually measurable cardinals, strategic $ω$-Ramsey cardinals and remarkable cardinals are equiconsistent.
- Every virtually measurable cardinal is strategic $ω$-Ramsey, and every strategic $ω$-Ramsey cardinal is virtually measurable in $L$.
- 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.
- Therefore, if $κ$ is a strategic ω-Ramsey cardinal then $L_κ \models \text{“there is a proper class of $ω$-Ramseys”}$.
Equiconsistency with the measurable cardinal:
- The existence of a strategic $(ω+1)$-Ramsey cardinal (and of strategic fully Ramsey cardinal) is equiconsistent with the existence of a measurable cardinal.[6]
- If $κ$ is a measurable cardinal, then $κ$ is $κ$-very Ramsey. If a cardinal is $ω_1$-very Ramsey (=strategic $ω_1$-Ramsey cardinal), then it is measurable in the core model unless $0^\P$ exists and an inner model with a Woodin cardinal exists.[4, 6]
- If $κ$ is uncountable, $κ = κ^{<κ}$ and $2^κ = κ^+$, then the following are equivalent:[5]
- $κ$ is measurable.
- $κ$ satisfies the $κ^+$-filter property.
- $κ$ satisfies the strategic $κ^+$-filter property.
- On the other hand, starting from a $κ^{++}$-tall cardinal $κ$, it is consistent that there is a cardinal $κ$ with the strategic $κ^+$-filter property, however $κ$ is not measurable.
Being downwards absolute to the core model:[6]
- If $0^\P$ does not exist:
- If $α$ is a limit ordinal with uncountable cofinality, then being $α$-Ramsey is downwards absolute to $\mathbf{K}$.
- If $α$ is a limit ordinal, then genuine $α$-Ramseyness and normal $α$-Ramseyness are both downwards absolute to $\mathbf{K}$.
- if $α$ is a limit of limit ordinals, then $<α$-Ramseyness is downwards absolute to $\mathbf{K}$.
Strategic $α$-Ramsey (including coherent $<α$-Ramsey) and $α$-very Ramsey:[6]
- For limit ordinal $α$, every coherent $<ωα$-Ramsey is $ωα$-very Ramsey.
- For any ordinal $α$, every coherent $<α$-very Ramsey is coherent $<α$-Ramsey.
- For limit ordinal $α$, $κ$ is $ωα$-very Ramsey iff it is coherent $<ωα$-Ramsey.
- $κ$ is $λ$-very Ramsey iff it is strategic $λ$-Ramsey for any $λ$ with uncountable cofinality.
Hierarchy:[5]
- If $ω ≤ α_0 < α_1 ≤ κ$, both $α_0$ and $α_1$ are cardinals, and $κ$ is $α_1$-Ramsey, then there is a proper class of $α_0$-Ramsey cardinals in $V_κ$. If $α_0$ is regular, then $κ$ is a limit of $α_0$-Ramsey cardinals.
- If $α ≤ κ$ are both cardinals and $κ$ is $α$-Ramsey, then $κ$ has a well-founded $α$-filter sequence.
- If $ω ≤ α < β ≤ κ$ are cardinals and $κ$ has a $β$-filter sequence, then there is a proper class of $α$-Ramsey cardinals in $V_κ$. If $α$ is regular, then $κ$ is a limit of $α$-Ramsey cardinals.
Other:
- Every $(ω+1)$-Ramsey cardinal is a completely Ramsey stationary limit of completely Ramsey cardinals.[6]
- Every strongly Ramsey cardinal is a stationary limit of almost fully Ramseys.[6]
- Fully Ramsey cardinals are super Ramsey limits of super Ramsey cardinals.[5]
- Measurable cardinals are limits of fully Ramsey cardinals.[5]
$(α, β)$-Ramsey cardinals
(All information from [6])
$κ$ is $(α, β)$-Ramsey iff player I has no winning strategy in $\mathcal{G}^θ_α(κ, β)$ for all regular $θ > κ$.
Of course, this notion is interesting only for $\mathrm{cof}(α) = ω$.
$α$-Ramsey cardinals are by definition equivalent to $(α, 0)$-Ramsey cardinals.
Position in the hierarchy of Erdős and iterable cardinals:
- For $β > 0$, every $(α, β)$-Ramsey is a $β$-iterable stationary limit of $β$-iterables.
- For additively closed $ω \leq α \leq ω_1$, any $α$-Erdős cardinal is a limit of $(ω, α)$-Ramsey cardinals.
This means also that $(ω, α)$-Ramsey cardinals are consistent with $V = L$ if $α < ω_1^L$ and that they are not if $α = ω_1$ .
$(γ, θ)$-Ramsey cardinals
$κ$ is $(γ, θ)$-Ramsey iff player I has no winning strategy in $\mathcal{G}^θ_γ(κ)$ (i.e. $κ$ is $γ$-Ramsey iff it is $(γ, θ)$-Ramsey for every $θ > κ$). Not much is known about them in general.[6]
M-rank
(from [16])
M-rank for Ramsey and Ramsey-like cardinals is analogous to Mitchell rank. A difference is that M-rank for Ramsey-like cardinals can be at most $\kappa^+$ (because an ultrapower of a weak $κ$-model has size at most $κ$) and Mitchell rank for measurable cardinals can be at most $(2^\kappa)^+$.
Definition of the M-order: For $κ$ having a large-cardinal property $\mathscr{P}$ with an embedding characterisation and for two witness collections $\mathcal{U}$ and $\mathcal{W}$ of $\mathscr{P}$-measures, we say that $U⊳W$ if
- for every $W∈\mathcal{W}$ and $A⊆κ$ in the ultrapower $N_W$ of $M_W$ by $W$, there is an $A$-good $U∈ \mathcal{U} ∩ N_W$ such that $N_W \models \text{“$\mathcal{U}$ is an $A$-good $\mathscr{P}$-measure on $κ$”}$.
- $\mathcal{U} ⊆ ⋃_{W∈\mathcal{W}} N_W$.
Results:
- Any strongly Ramsey cardinal $κ$ has Ramsey M-rank $κ^+$,
- any super Ramsey cardinal $κ$ has strongly Ramsey M-rank $κ^+$
- and any measurable cardinal $κ$ has super Ramsey M-rank $κ^+$.
Ramsey and Ramsey-like M-orders can be softly killed (Rank $α$ can be turned into rank $β$ for any $β < α$) using cofinality-preserving forcing extension.
References
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