ZFC set theory, which includes the axiom of choice, implies that every infinite set has an aleph number as its cardinality ( i . e . is equinumerous with its initial ordinal ), and thus the initial ordinals of the aleph numbers serve as a class of representatives for all possible infinite cardinal numbers.
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A finite sequence of finite ordinals always has a finite maximum, so \ omega cannot be the limit of any sequence of type less than \ omega whose elements are ordinals less than \ omega, and is therefore a regular ordinal . \ aleph _ 0 ( aleph-null ) is a regular cardinal because its initial ordinal, \ omega, is regular.
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Interestingly, if 0 # holds in V, then there is a closed unbounded class of ordinals that are indiscernible in L . While some of these are not even initial ordinals in V, they have all the large cardinal properties weaker than 0 # in L . Furthermore, any strictly increasing class function from the class of indiscernibles to itself can be extended in a unique way to an elementary embedding of L into L . This gives L a nice structure of repeating segments.
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