### Initial ordinal

**
The ****von Neumann cardinal assignment** is a cardinal assignment which uses ordinal numbers. For a well-ordered set *U*, we define its cardinal number to be the smallest ordinal number equinumerous to *U*. More precisely:

- $|U|\; =\; \backslash mathrm\{card\}(U)\; =\; \backslash inf\; \backslash \{\; \backslash alpha\; \backslash in\; ON\; \backslash \; |\backslash \; \backslash alpha\; =\_c\; U\; \backslash \},$

where ON is the class of ordinals. This ordinal is also called the **initial ordinal** of the cardinal.

That such an ordinal exists and is unique is guaranteed by the fact that *U* is well-orderable and that the class of ordinals is well-ordered, using the axiom of replacement. With the full axiom of choice, every set is well-orderable, so every set has a cardinal; we order the cardinals using the inherited ordering from the ordinal numbers. This is readily found to coincide with the ordering via ≤_{c}. This is a well-ordering of cardinal numbers.

## Initial ordinal of a cardinal

Each ordinal has an associated cardinal, its cardinality, obtained by simply forgetting the order. Any well-ordered set having that ordinal as its order type has the same cardinality. The smallest ordinal having a given cardinal as its cardinality is called the initial ordinal of that cardinal. 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 initial ordinal *is* a cardinal.

The α-th infinite initial ordinal is written $\backslash omega\_\backslash alpha$. Its cardinality is written ℵ_{α} (the α-th aleph number). For example, the cardinality of ω_{0} = ω is ℵ_{0}, which is also the cardinality of ω^{2}, ω^{ω}, and ε_{0} (all are countable ordinals). So (assuming the axiom of choice) we identify ω_{α} with ℵ_{α}, except that the notation ℵ_{α} is used for writing cardinals, and ω_{α} for writing ordinals. This is important because arithmetic on cardinals is different from arithmetic on ordinals, for example ℵ_{α}^{2} = ℵ_{α} whereas ω_{α}^{2} > ω_{α}. Also, ω_{1} is the smallest uncountable ordinal (to see that it exists, consider the set of equivalence classes of well-orderings of the natural numbers; each such well-ordering defines a countable ordinal, and ω_{1} is the order type of that set), ω_{2} is the smallest ordinal whose cardinality is greater than ℵ_{1}, and so on, and ω_{ω} is the limit of ω_{n} for natural numbers *n* (any limit of cardinals is a cardinal, so this limit is indeed the first cardinal after all the ω_{n}).

Infinite initial ordinals are limit ordinals. Using ordinal arithmetic, α < ω_{β} implies α+ω_{β} = ω_{β}, and 1 ≤ α < ω_{β} implies α·ω_{β} = ω_{β}, and 2 ≤ α < ω_{β} implies α^{ωβ} = ω_{β}. Using the Veblen hierarchy, β ≠ 0 and α < ω_{β} imply $\backslash varphi\_\{\backslash alpha\}(\backslash omega\_\{\backslash beta\})\; =\; \backslash omega\_\{\backslash beta\}\; \backslash ,$ and Γ_{ωβ} = ω_{β}. Indeed, one can go far beyond this. So as an ordinal, an infinite initial ordinal is an extremely strong kind of limit.

## See also

## References

- Y.N. Moschovakis
*Notes on Set Theory*(1994 Springer) p. 198