Ineffable cardinal

Kind of large cardinal number

In the mathematics of transfinite numbers, an ineffable cardinal is a certain kind of large cardinal number, introduced by Jensen & Kunen (1969). In the following definitions,  κ {\displaystyle \kappa } will always be a regular uncountable cardinal number.

A cardinal number κ {\displaystyle \kappa } is called almost ineffable if for every  f : κ → P ( κ ) {\displaystyle f:\kappa \to {\mathcal {P}}(\kappa )} (where  P ( κ ) {\displaystyle {\mathcal {P}}(\kappa )} is the powerset of  κ {\displaystyle \kappa } ) with the property that  f ( δ ) {\displaystyle f(\delta )} is a subset of  δ {\displaystyle \delta } for all ordinals  δ < κ {\displaystyle \delta <\kappa } , there is a subset  S {\displaystyle S} of  κ {\displaystyle \kappa } having cardinality  κ {\displaystyle \kappa } and homogeneous for  f {\displaystyle f} , in the sense that for any  δ 1 < δ 2 {\displaystyle \delta _{1}<\delta _{2}} in  S {\displaystyle S} ,  f ( δ 1 ) = f ( δ 2 ) ∩ δ 1 {\displaystyle f(\delta _{1})=f(\delta _{2})\cap \delta _{1}} .

A cardinal number κ {\displaystyle \kappa } is called ineffable if for every binary-valued function  f : [ κ ] 2 → { 0 , 1 } {\displaystyle f:[\kappa ]^{2}\to \\{0,1\\}} , there is a stationary subset of  κ {\displaystyle \kappa } on which  f {\displaystyle f} is homogeneous: that is, either  f {\displaystyle f} maps all unordered pairs of elements drawn from that subset to zero, or it maps all such unordered pairs to one. An equivalent formulation is that a cardinal  κ {\displaystyle \kappa } is ineffable if for every sequence ⟨ A α : α ∈ κ ⟩ {\displaystyle \langle A_{\alpha }:\alpha \in \kappa \rangle } such that each  A α ⊆ α {\displaystyle A_{\alpha }\subseteq \alpha } , there is  A ⊆ κ {\displaystyle A\subseteq \kappa } such that  { α ∈ κ : A ∩ α = A α } {\displaystyle \\{\alpha \in \kappa :A\cap \alpha =A_{\alpha }\\}} is stationary in κ.

Another equivalent formulation is that a regular uncountable cardinal  κ {\displaystyle \kappa } is ineffable if for every set  S {\displaystyle S} of cardinality  κ {\displaystyle \kappa } of subsets of  κ {\displaystyle \kappa } , there is a normal (i.e. closed under diagonal intersection) non-trivial  κ {\displaystyle \kappa } -complete filter F {\displaystyle {\mathcal {F}}} on  κ {\displaystyle \kappa } deciding  S {\displaystyle S} : that is, for any  X ∈ S {\displaystyle X\in S} , either  X ∈ F {\displaystyle X\in {\mathcal {F}}} or  κ ∖ X ∈ F {\displaystyle \kappa \setminus X\in {\mathcal {F}}} .[1] This is similar to a characterization of weakly compact cardinals.

More generally,  κ {\displaystyle \kappa } is called  n {\displaystyle n} -ineffable (for a positive integer  n {\displaystyle n} ) if for every  f : [ κ ] n → { 0 , 1 } {\displaystyle f:[\kappa ]^{n}\to \\{0,1\\}} there is a stationary subset of  κ {\displaystyle \kappa } on which  f {\displaystyle f} is  n {\displaystyle n} -homogeneous (takes the same value for all unordered  n {\displaystyle n} -tuples drawn from the subset). Thus, it is ineffable if and only if it is 2-ineffable. Ineffability is strictly weaker than 3-ineffability.[2]p. 399

A totally ineffable cardinal is a cardinal that is  n {\displaystyle n} -ineffable for every  2 ≤ n < ℵ 0 {\displaystyle 2\leq n<\aleph _{0}} . If  κ {\displaystyle \kappa } is  ( n + 1 ) {\displaystyle (n+1)} -ineffable, then the set of  n {\displaystyle n} -ineffable cardinals below  κ {\displaystyle \kappa } is a stationary subset of  κ {\displaystyle \kappa } .

Every  n {\displaystyle n} -ineffable cardinal is  n {\displaystyle n} -almost ineffable (with set of  n {\displaystyle n} -almost ineffable below it stationary), and every  n {\displaystyle n} -almost ineffable is  n {\displaystyle n} -subtle (with set of  n {\displaystyle n} -subtle below it stationary). The least  n {\displaystyle n} -subtle cardinal is not even weakly compact (and unlike ineffable cardinals, the least  n {\displaystyle n} -almost ineffable is  Π 2 1 {\displaystyle \Pi _{2}^{1}} -describable), but  ( n − 1 ) {\displaystyle (n-1)} -ineffable cardinals are stationary below every  n {\displaystyle n} -subtle cardinal.

A cardinal κ is completely ineffable if there is a non-empty  R ⊆ P ( κ ) {\displaystyle R\subseteq {\mathcal {P}}(\kappa )} such that
\- every  A ∈ R {\displaystyle A\in R} is stationary
\- for every  A ∈ R {\displaystyle A\in R} and  f : [ κ ] 2 → { 0 , 1 } {\displaystyle f:[\kappa ]^{2}\to \\{0,1\\}} , there is  B ⊆ A {\displaystyle B\subseteq A} homogeneous for f with  B ∈ R {\displaystyle B\in R} .

Using any finite  n {\displaystyle n}  > 1 in place of 2 would lead to the same definition, so completely ineffable cardinals are totally ineffable (and have greater consistency strength). Completely ineffable cardinals are  Π n 1 {\displaystyle \Pi _{n}^{1}} -indescribable for every n, but the property of being completely ineffable is  Δ 1 2 {\displaystyle \Delta _{1}^{2}} .

The consistency strength of completely ineffable is below that of 1-iterable cardinals, which in turn is below remarkable cardinals, which in turn is below ω-Erdős cardinals. A list of large cardinal axioms by consistency strength is available in the section below.

## See also

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* List of large cardinal properties

## References

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* Friedman, Harvey (2001), "Subtle cardinals and linear orderings", Annals of Pure and Applied Logic, 107 (1–3): 1–34, doi:10.1016/S0168-0072(00)00019-11–3&rft.pages=1-34&rft.date=2001&rft_id=info:doi/10.1016/S0168-0072(00)00019-1&rft.aulast=Friedman&rft.aufirst=Harvey&rfr_id=info:sid/en.wikipedia.org:Ineffable+cardinal" class="Z3988">.
* Jensen, Ronald; Kunen, Kenneth (1969), Some Combinatorial Properties of L and V, Unpublished manuscript

### Citations

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1. ^ Holy, Peter; Schlicht, Philipp (2017). "A hierarchy of Ramsey-like cardinals". arXiv:1710.10043 [math.LO].
2. ^ K. Kunen,. "Combinatorics". In Handbook of Mathematical Logic, Studies in Logic and the Foundations of mathematical vol. 90, ed. J. Barwise (1977)