The least ordinal associated with a given cardinal is called the " initial ordinal " of that cardinal.
12.
Any limit of regular ordinals is a limit of initial ordinals and thus is also initial but need not be regular.
13.
Any limit of regular ordinals is a limit of initial ordinals and thus is also initial even if it is not regular, which it usually is not.
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The axiom of choice is equivalent to the statement that every set can be well-ordered, i . e . that every cardinal has an initial ordinal.
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In theories with the axiom of choice, the cardinal number of any set has an initial ordinal, and one may employ the Von Neumann cardinal assignment as the cardinal's representation.
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If that initial ordinal is \ omega _ { \ lambda } \,, then the cardinal number is of the form \ aleph _ \ lambda for the same ordinal subscript & lambda;.
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Then we can apply the axiom of replacement to replace each element of that powerset of " x " by the initial ordinal number of the same cardinality ( or zero, if there is no such ordinal ).
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In this case, the ordinals 0, 1, \ omega, \ omega _ 1, and \ omega _ 2 are regular, whereas 2, 3, \ omega _ \ omega, and ? ?? are initial ordinals which are not regular.
19.
In this case, the ordinals 0, 1, \ omega, \ omega _ 1, and \ omega _ 2 are regular, whereas 2, 3, \ omega _ \ omega, and ? ?? are initial ordinals that are not regular.
20.
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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