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An infinite ordinal is a regular ordinal if it is a limit ordinal that is not the limit of a set of smaller ordinals that as a set has order type less than . A regular ordinal is always an initial ordinal , though some initial ordinals are not regular, e.g., ω ω {\displaystyle \omega _{\omega }} (see the example below).
So ordinal numbers exist and are essentially unique. Ordinal numbers are distinct from cardinal numbers, which measure the size of sets. Although the distinction between ordinals and cardinals is not always apparent on finite sets (one can go from one to the other just by counting labels), they are very different in the infinite case, where ...
Every finite ordinal (natural number) is initial, but most infinite ordinals are not initial. 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. In this case, it is traditional to identify the cardinal number with its initial ordinal, and we say that the ...
The definition of a finite set is given independently of natural numbers: [3] Definition: A set is finite if and only if any non empty family of its subsets has a minimal element for the inclusion order. Definition: a cardinal n is a natural number if and only if there exists a finite set of which the cardinal is n. 0 = Card (∅)
For each ordinal, + is the least cardinal number greater than . The cardinality of the natural numbers is denoted aleph-null ( ℵ 0 {\displaystyle \aleph _{0}} ), while the cardinality of the real numbers is denoted by " c {\displaystyle {\mathfrak {c}}} " (a lowercase fraktur script "c"), and is also referred to as the cardinality of the ...
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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