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:
“But those rare souls whose spirit gets magically into the hearts of men, leave behind them something more real and warmly personal than bodily presence, an ineffable and eternal thing. It is everlasting life touching us as something more than a vague, recondite concept. The sound of a great name dies like an echo; the splendor of fame fades into nothing; but the grace of a fine spirit pervades the places through which it has passed, like the haunting loveliness of mignonette.”
—James Thurber (1894–1961)
“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)