Fixed Points of Omega
For any ordinal α we have
In many cases is strictly greater than α. For example, for any successor ordinal α this holds. There are, however, some limit ordinals which are fixed points of the omega function, because of the fixed-point lemma for normal functions. The first such is the limit of the sequence
Any weakly inaccessible cardinal is also a fixed point of the aleph function. This can be shown in ZFC by noting that if is a weakly inaccessible cardinal then for every initial ordinal of a cardinal smaller than which is not the itself a weakly inaccessible cardinal, is different than : either is a successor ordinal (in which case is a successor cardinal and hence not weakly inaccessible) or is a limit ordinal whose cofinality is smaller than its associated cardinal (in which case is larger than its cofinality and hence not weakly inaccessible). Thus, if there were weakly inaccessible cardinals which are not fixed points of the function, the smallest of these would not be the image of any ordinal by the function.
Read more about this topic: Aleph Number
Famous quotes containing the words fixed and/or points:
“The foot of the heavenly ladder, which we have got to mount in order to reach the higher regions, has to be fixed firmly in every-day life, so that everybody may be able to climb up it along with us. When people then find that they have got climbed up higher and higher into a marvelous, magical world, they will feel that that realm, too, belongs to their ordinary, every-day life, and is, merely, the wonderful and most glorious part thereof.”
—E.T.A.W. (Ernst Theodor Amadeus Wilhelm)
“Wi’ joy unfeigned brothers and sisters meet,
An’ each for other’s weelfare kindly spiers:
The social hours, swift-winged, unnoticed fleet;
Each tells the uncos that he sees or hears;
The parents, partial, eye their hopeful years;
Anticipation forward points the view:”
—Robert Burns (1759–1796)