Aleph Number - Fixed Points of Omega

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.