ineffable cardinal

{{short description|Kind of large cardinal number}}

In the mathematics of transfinite numbers, an ineffable cardinal is a certain kind of large cardinal number, introduced by {{harvtxt|Jensen|Kunen|1969}}. In the following definitions, $\backslash kappa$ will always be a regular uncountable cardinal number.

A cardinal number $\backslash kappa$ is called almost ineffable if for every $f:\; \backslash kappa\; \backslash to\; \backslash mathcal\{P\}(\backslash kappa)$ (where $\backslash mathcal\{P\}(\backslash kappa)$ is the powerset of $\backslash kappa$) with the property that $f(\backslash delta)$ is a subset of $\backslash delta$ for all ordinals , there is a subset $S$ of $\backslash kappa$ having cardinality $\backslash kappa$ and homogeneous for $f$, in the sense that for any in $S$, $f(\backslash delta\_1)\; =\; f(\backslash delta\_2)\; \backslash cap\; \backslash delta\_1$.

A cardinal number $\backslash kappa$ is called ineffable if for every binary-valued function $f\; :\; [\backslash kappa]^2\backslash to\; \backslash \{0,1\backslash \}$, there is a stationary subset of $\backslash kappa$ on which $f$ is homogeneous: that is, either $f$ 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 $\backslash kappa$ is ineffable if for every sequence $\backslash langle\; A\_\backslash alpha:\backslash alpha\backslash in\backslash kappa\backslash rangle$ such that each $A\_\backslash alpha\backslash subseteq\backslash alpha$,

there is $A\backslash subseteq\backslash kappa$ such that $\backslash \{\backslash alpha\backslash in\backslash kappa:A\backslash cap\backslash alpha=A\_\backslash alpha\backslash \}$ is stationary in {{math|κ}}.

Another equivalent formulation is that a regular uncountable cardinal $\backslash kappa$ is ineffable if for every set $S$ of cardinality $\backslash kappa$ of subsets of $\backslash kappa$, there is a normal (i.e. closed under diagonal intersection) non-trivial $\backslash kappa$-complete filter $\backslash mathcal\; F$ on $\backslash kappa$ deciding $S$: that is, for any $X\backslash in\; S$, either $X\backslash in\backslash mathcal\; F$ or $\backslash kappa\backslash setminus\; X\backslash in\backslash mathcal\; F$.{{cite arXiv|eprint=1710.10043 |last1=Holy |first1=Peter |last2=Schlicht |first2=Philipp |title=A hierarchy of Ramsey-like cardinals |date=2017 |class=math.LO }} This is similar to a characterization of weakly compact cardinals.

More generally, $\backslash kappa$ is called $n$-ineffable (for a positive integer $n$) if for every $f\; :\; [\backslash kappa]^n\backslash to\; \backslash \{0,1\backslash \}$ there is a stationary subset of $\backslash kappa$ on which $f$ is $n$-homogeneous (takes the same value for all unordered $n$-tuples drawn from the subset). Thus, it is ineffable if and only if it is 2-ineffable. Ineffability is strictly weaker than 3-ineffability.K. Kunen,. "Combinatorics". In Handbook of Mathematical Logic, Studies in Logic and the Foundations of mathematical vol. 90, ed. J. Barwise (1977)^{p. 399}

A totally ineffable cardinal is a cardinal that is $n$-ineffable for every . If $\backslash kappa$ is $(n+1)$-ineffable, then the set of $n$-ineffable cardinals below $\backslash kappa$ is a stationary subset of $\backslash kappa$.

Every $n$-ineffable cardinal is $n$-almost ineffable (with set of $n$-almost ineffable below it stationary), and every $n$-almost ineffable is $n$-subtle (with set of $n$-subtle below it stationary). The least $n$-subtle cardinal is not even weakly compact (and unlike ineffable cardinals, the least $n$-almost ineffable is $\backslash Pi^1\_2$-describable), but $(n-1)$-ineffable cardinals are stationary below every $n$-subtle cardinal.

A cardinal κ is completely ineffable if there is a non-empty $R\; \backslash subseteq\; \backslash mathcal\{P\}(\backslash kappa)$ such that

- every $A\; \backslash in\; R$ is stationary

- for every $A\; \backslash in\; R$ and $f\; :\; [\backslash kappa]^2\backslash to\; \backslash \{0,1\backslash \}$, there is $B\; \backslash subseteq\; A$ homogeneous for f with $B\; \backslash in\; R$.

Using any finite $n$ > 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 $\backslash Pi^1\_n$-indescribable for every n, but the property of being completely ineffable is $\backslash Delta^2\_1$.

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.