In the mathematics of transfinite numbers, an ineffable cardinal is a certain kind of large cardinal number, introduced by Jensen & Kunen (1969).
A cardinal number is called almost ineffable if for every (where is the powerset of ) with the property that is a subset of for all ordinals, there is a subset of having cardinal and homogeneous for, in the sense that for any in, .
A cardinal number is called ineffable if for every binary-valued function, there is a stationary subset of on which is homogeneous: that is, either maps all unordered pairs of elements drawn from that subset to zero, or it maps all such unordered pairs to one.
More generally, is called -ineffable (for a positive integer ) if for every there is a stationary subset of on which is -homogeneous (takes the same value for all unordered -tuples drawn from the subset). Thus, it is ineffable if and only if it is 2-ineffable.
A totally ineffable cardinal is a cardinal that is -ineffable for every . If is -ineffable, then the set of -ineffable cardinals below is a stationary subset of .
Totally ineffable cardinals are of greater consistency strength than subtle cardinals and of lesser consistency strength than remarkable cardinals. A list of large cardinal axioms by consistency strength is available here.
Famous quotes containing the words ineffable and/or cardinal:
“I have written a wicked book, and feel spotless as the lamb. Ineffable socialities are in me. I would sit down and dine with you and all the gods in old Rome’s Pantheon. It is a strange feeling—no hopefulness is in it, no despair. Content—that is it; and irresponsibility; but without licentious inclination.”
—Herman Melville (1819–1891)
“Honest towards ourselves and towards anyone else who is our friend; brave towards the enemy; magnanimous towards the defeated; polite—always: this is how the four cardinal virtues want us to act.”
—Friedrich Nietzsche (1844–1900)