Ineffable cardinal
Ineffable cardinals were introduced by Jensen and Kunen in [1] and arose out of their study of $\diamondsuit$ principles. An uncountable regular cardinal $\kappa$ is ineffable if for every sequence $\langle A_\alpha\mid \alpha<\kappa\rangle$ with $A_\alpha\subseteq \alpha$ there is $A\subseteq\kappa$ such that the set $S=\{\alpha<\kappa\mid A\cap \alpha=A_\alpha\}$ is stationary. Equivalently an uncountable regular $\kappa$ is ineffable if and only if for every function $F:[\kappa]^2\rightarrow 2$ there is a stationary $H\subseteq\kappa$ such that $F\upharpoonright [H]^2$ is constant [1]. This second characterization strengthens a characterization of weakly compact cardinals which requires that there exist such an $H$ of size $\kappa$.
If $\kappa$ is ineffable, then $\diamondsuit_\kappa$ holds and there cannot be a slim $\kappa$-Kurepa tree [1] . A $\kappa$-Kurepa tree is a tree of height $\kappa$ having levels of size less than $\kappa$ and at least $\kappa^+$-many branches. A $\kappa$-Kurepa tree is slim if every infinite level $\alpha$ has size at most $|\alpha|$.
Uncountable regular $\kappa$ is ineffable iff whenever $(M_\gamma)_{\gamma<\kappa}$ is a $\kappa$-sequence of models of a language with fewer than $\kappa$ symbols and $M_\lambda=\bigcup_{\lambda'<\lambda}<_{\lambda'}$ for limit $\lambda$, there is a stationary set $X\subseteq\kappa$ such that $\alpha,\beta\in X\land\alpha<\beta\rightarrow M_\alpha\prec M_\beta$. (Jensen, "Some combinatorial properties of L and V", notes, 2010, p.18)
An uncountable cardinal κ has the normal filter property iff it is ineffable.[2]
Contents
Ineffable cardinals and the constructible universe
Ineffable cardinals are downward absolute to $L$. In $L$, an inaccessible cardinal $\kappa$ is ineffable if and only if there are no slim $\kappa$-Kurepa trees. Thus, for inaccessible cardinals, in $L$, ineffability is completely characterized using slim Kurepa trees. [1]
If $0^\sharp$ exists, then every Silver indiscernible is ineffable in $L$. [3]
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. The least strongly Ramsey cardinal also is not ineffable, but super weakly Ramsey cardinals are ineffable. $1$-iterable (=weakly Ramsey) cardinals are weakly ineffable and stationary limits of completely ineffable cardinals. The least $1$-iterable cardinal is not ineffable. [2, 4]
Relations with other large cardinals
- Measurable cardinals are ineffable and stationary limits of ineffable cardinals.
- $\omega$-Erdős cardinals are stationary limits of ineffable cardinals, but not ineffable since they are $\Pi_1^1$-describable. [3]
- Ineffable cardinals are $\Pi^1_2$-indescribable [1].
- Ineffable cardinals are limits of totally indescribable cardinals. [1] ([5] for proof)
- For a cardinal $κ=κ^{<κ}$, $κ$ is ineffable iff it is normal 0-Ramsey. [6]
Weakly ineffable cardinal
Weakly ineffable cardinals (also called almost ineffable) were introduced by Jensen and Kunen in [1] as a weakening of ineffable cardinals. An uncountable regular cardinal $\kappa$ is weakly ineffable if for every sequence $\langle A_\alpha\mid \alpha<\kappa\rangle$ with $A_\alpha\subseteq \alpha$ there is $A\subseteq\kappa$ such that the set $S=\{\alpha<\kappa\mid A\cap \alpha=A_\alpha\}$ has size $\kappa$. If $\kappa$ is weakly ineffable, then $\diamondsuit_\kappa$ holds.
- Weakly ineffable cardinals are downward absolute to $L$. [1]
- Weakly ineffable cardinals are $\Pi_1^1$-indescribable. [1]
- Ineffable cardinals are limits of weakly ineffable cardinals.
- Weakly ineffable cardinals are limits of totally indescribable cardinals. [1] ([5] for proof)
- For a cardinal $κ=κ^{<κ}$, $κ$ is weakly ineffable iff it is genuine 0-Ramsey. [6]
Subtle cardinal
Subtle cardinals were introduced by Jensen and Kunen in [1] as a weakening of weakly ineffable cardinals. A uncountable regular cardinal $\kappa$ is subtle if for every for every $\langle A_\alpha\mid \alpha<\kappa\rangle$ with $A_\alpha\subseteq \alpha$ and every closed unbounded $C\subseteq\kappa$ there are $\alpha<\beta$ in $C$ such that $A_\beta\cap\alpha=A_\alpha$. If $\kappa$ is subtle, then $\diamondsuit_\kappa$ holds.
- Subtle cardinals are downward absolute to $L$. [1]
- Weakly ineffable cardinals are limits of subtle cardinals. [1]
- Subtle cardinals are stationary limits of totally indescribable cardinals. [1, 7]
- The least subtle cardinal is not weakly compact as it is $\Pi_1^1$-describable.
- $\alpha$-Erdős cardinals are subtle. [1]
- If $δ$ is a subtle cardinal,
- the set of cardinals $κ$ below $δ$ that are strongly uplifting in $V_δ$ is stationary.[8]
- for every class $\mathcal{A}$, in every club $B ⊆ δ$ there is $κ$ such that $\langle V_δ, \mathcal{A} ∩ V_δ \rangle \models \text{“$κ$ is $\mathcal{A}$-shrewd.”}$.[9] (The set of cardinals $κ$ below $δ$ that are $\mathcal{A}$-shrewd in $V_δ$ is stationary.)
- there is an $\eta$-shrewd cardinal below $δ$ for all $\eta < δ$.[9]
Ethereal cardinal
Ethereal cardinals were introduced by Ketonen in [10] (information in this section from there) as a weakening of subtle cardinals.
Definition:
- A regular cardinal $κ$ is called ethereal if for every club $C$ in $κ$ and sequence $(S_α|α < κ)$ of sets such that for $α < κ$, $|S_α| = |α|$ and $S_α ⊆ α$, there are elements $α, β ∈ C$ such that $α < β$ and $|S_α ∩ S_β| = |α|$. I.e., symbolically(?):
$$κ \text{ – ethereal} \overset{\text{def}}{⟺} \left( κ \text{ – regular} ∧ \left( \forall_{C \text{ – club in $κ$}} \forall_{S : κ → \mathcal{P}(κ)} \left( \forall_{α < κ} |S_α| = |α| ∧ S_α ⊆ α \right) ⟹ \left( \exists_{α, β ∈ C} α < β ∧ |S_α ∩ S_β| = |α| \right) \right) \right)$$
Properties:
- Every subtle cardinal is obviously ethereal.
- Every ethereal cardinal is weakly inaccessible.
- A strongly inaccessible cardinal is ethereal if and only if it is subtle.
- If $κ$ is ethereal and $2^\underset{\smile}{κ} = κ$, then $♢_κ$ holds (where $2^\underset{\smile}{κ} = \bigcup \{ 2^α | α < κ \}$ is the weak power of $κ$).
To be expanded.
$n$-ineffable cardinal
The $n$-ineffable cardinals for $2\leq n<\omega$ were introduced by Baumgartner in [11] as a strengthening of ineffable cardinals. A cardinal is $n$-ineffable if for every function $F:[\kappa]^n\rightarrow 2$ there is a stationary $H\subseteq\kappa$ such that $F\upharpoonright [H]^n$ is constant.
- $2$-ineffable cardinals are exactly the ineffable cardinals.
- an $n+1$-ineffable cardinal is a stationary limit of $n$-ineffable cardinals. [11]
A cardinal $\kappa$ is totally ineffable if it is $n$-ineffable for every $n$.
- a $1$-iterable cardinal is a stationary limit of totally ineffable cardinals. (this follows from material in [4])
- For each particular natural number in the metatheory $n ≥ 1$, $\mathrm{ZFC} + \mathrm{BTEE}$ proves that the critical point of $j$ is $n$-ineffable.[12]
Helix
(Information in this subsection come from [7] unless noted otherwise.)
For $k \geq 1$ we define:
- $\mathcal{P}(x)$ 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.
- $f:\mathcal{P}_k(\lambda) \to \mathcal{P}(\lambda)$ is regressive iff for all $A \in \mathcal{P}_k(\lambda)$, we have $f(A) \subseteq \min(A)$.
- $E$ is $f$-homogenous iff $E \subseteq \lambda$ and for all $B,C \in \mathcal{P}_k(E)$, we have $f(B) \cap \min(B \cup C) = f(C) \cap \min(B \cup C)$.
- $\lambda$ is $k$-subtle iff $\lambda$ is a limit ordinal and for all clubs $C \subseteq \lambda$ and regressive $f:\mathcal{P}_k(\lambda) \to \mathcal{P}(\lambda)$, there exists an $f$-homogenous $A \in \mathcal{P}_{k+1}(C)$.
- $\lambda$ is $k$-almost ineffable iff $\lambda$ is a limit ordinal and for all regressive $f:\mathcal{P}_k(\lambda) \to \mathcal{P}(\lambda)$, there exists an $f$-homogenous $A \subseteq \lambda$ of cardinality $\lambda$.
- $\lambda$ is $k$-ineffable iff $\lambda$ is a limit ordinal and for all regressive $f:\mathcal{P}_k(\lambda) \to \mathcal{P}(\lambda)$, there exists an $f$-homogenous stationary $A \subseteq \lambda$.
$0$-subtle, $0$-almost ineffable and $0$-ineffable cardinals can be defined as “uncountable regular cardinals” because for $k \geq 1$ all three properties imply being uncountable regular cardinals.
- For $k \geq 1$, if $\kappa$ is a $k$-ineffable cardinal, then $\kappa$ is $k$-almost ineffable and the set of $k$-almost ineffable cardinals is stationary in $\kappa$.
- For $k \geq 1$, if $\kappa$ is a $k$-almost ineffable cardinal, then $\kappa$ is $k$-subtle and the set of $k$-subtle cardinals is stationary in $\kappa$.
- For $k \geq 1$, if $\kappa$ is a $k$-subtle cardinal, then the set of $(k-1)$-ineffable cardinals is stationary in $\kappa$.
- For $k \geq n \geq 0$, all $k$-ineffable cardinals are $n$-ineffable, all $k$-almost ineffable cardinals are $n$-almost ineffable and all $k$-subtle cardinals are $n$-subtle.
This structure is similar to the double helix of $n$-fold variants and earlier known although smaller.[13]
Completely ineffable cardinal
Completely ineffable cardinals were introduced in [5] as a strengthening of ineffable cardinals. Define that a collection $R\subseteq P(\kappa)$ is a stationary class if
- $R\neq\emptyset$,
- for all $A\in R$, $A$ is stationary in $\kappa$,
- if $A\in R$ and $B\supseteq A$, then $B\in R$.
A cardinal $\kappa$ is completely ineffable if there is a stationary class $R$ such that for every $A\in R$ and $F:[A]^2\to2$, there is $H\in R$ such that $F\upharpoonright [H]^2$ is constant.
Relations:
- Completely ineffable cardinals are downward absolute to $L$. [5]
- Completely ineffable cardinals are limits of ineffable cardinals. [5]
- There are stationarily many completely ineffable, greatly Erdős cardinals below any Ramsey cardinal.[14]
- The following are equivalent:[6]
- $κ$ is completely ineffable.
- $κ$ is coherent $<ω$-Ramsey.
- $κ$ has the $ω$-filter property.
- Every completely ineffable is a stationary limit of $<ω$-Ramseys.[6]
- Completely Ramsey cardinals and $ω$-Ramsey cardinals are completely ineffable.[6]
- $ω$-Ramsey cardinals are limits of completely ineffable cardinals.[2]
- Consistency of a completely ineffable cardinal implies consistency of the theory $\mathrm{BTEE}$ (Basic Theory of Elementary Embeddings).[1]
References
- Jensen, Ronald and Kunen, Kenneth. Some combinatorial properties of $L$ and $V$. Unpublished, 1969. www bibtex
- Holy, Peter and Schlicht, Philipp. A hierarchy of Ramsey-like cardinals. Fundamenta Mathematicae 242:49-74, 2018. www arχiv DOI bibtex
- Jech, Thomas J. Set Theory. Third, Springer-Verlag, Berlin, 2003. (The third millennium edition, revised and expanded) www bibtex
- Gitman, Victoria. Ramsey-like cardinals. The Journal of Symbolic Logic 76(2):519-540, 2011. www arχiv MR bibtex
- Abramson, Fred and Harrington, Leo and Kleinberg, Eugene and Zwicker, William. Flipping properties: a unifying thread in the theory of large cardinals. Ann Math Logic 12(1):25--58, 1977. MR bibtex
- Nielsen, Dan Saattrup and Welch, Philip. Games and Ramsey-like cardinals. , 2018. arχiv bibtex
- Friedman, Harvey M. Subtle cardinals and linear orderings. , 1998. www bibtex
- Hamkins, Joel David and Johnstone, Thomas A. Strongly uplifting cardinals and the boldface resurrection axioms. , 2014. arχiv bibtex
- Rathjen, Michael. The art of ordinal analysis. , 2006. www bibtex
- Ketonen, Jussi. Some combinatorial principles. Trans Amer Math Soc 188:387-394, 1974. DOI bibtex
- Baumgartner, James. Ineffability properties of cardinals. I. Infinite and finite sets (Colloq., Keszthely, 1973; dedicated to P. Erdős on his 60th birthday), Vol. I, pp. 109--130. Colloq. Math. Soc. János Bolyai, Vol. 10, Amsterdam, 1975. MR bibtex
- Corazza, Paul. The spectrum of elementary embeddings $j : V \to V$. Annals of Pure and Applied Logic 139(1--3):327-399, May, 2006. DOI bibtex
- Kentaro, Sato. Double helix in large large cardinals and iteration of elementary embeddings. Annals of Pure and Applied Logic 146(2-3):199-236, May, 2007. www DOI bibtex
- Sharpe, Ian and Welch, Philip. Greatly Erdős cardinals with some generalizations to the Chang and Ramsey properties. Ann Pure Appl Logic 162(11):863--902, 2011. www DOI MR bibtex