Von Neumann cardinal assignment

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 ${\displaystyle \alpha }$-th infinite initial ordinal is written ${\displaystyle \omega _{\alpha }}$. Its cardinality is written ${\displaystyle \aleph _{\alpha }}$ (the ${\displaystyle \alpha }$-th aleph number). For example, the cardinality of ${\displaystyle \omega _{0}=\omega }$ is ${\displaystyle \aleph _{0}}$, which is also the cardinality of ${\displaystyle \omega ^{2}}$, ${\displaystyle \omega ^{\omega }}$, and ${\displaystyle \epsilon _{0}}$ (all are countable ordinals). So we identify ${\displaystyle \omega _{\alpha }}$ with ${\displaystyle \aleph _{\alpha }}$, except that the notation ${\displaystyle \aleph _{\alpha }}$ is used for writing cardinals, and ${\displaystyle \omega _{\alpha }}$ for writing ordinals. This is important because arithmetic on cardinals is different from arithmetic on ordinals, for example ${\displaystyle \aleph _{\alpha }^{2}}$ = ${\displaystyle \aleph _{\alpha }}$ whereas ${\displaystyle \omega _{\alpha }^{2}}$ > ${\displaystyle \omega _{\alpha }}$. Also, ${\displaystyle \omega _{1}}$ 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 ${\displaystyle \omega _{1}}$ is the order type of that set), ${\displaystyle \omega _{2}}$ is the smallest ordinal whose cardinality is greater than ${\displaystyle \aleph _{1}}$, and so on, and ${\displaystyle \omega _{\omega }}$ is the limit of ${\displaystyle \omega _{n}}$ for natural numbers ${\displaystyle n}$ (any limit of cardinals is a cardinal, so this limit is indeed the first cardinal after all the ${\displaystyle \omega _{n}}$).

Infinite initial ordinals are limit ordinals. Using ordinal arithmetic, ${\displaystyle \alpha <\omega _{\beta }}$ implies ${\displaystyle \alpha +\omega _{\beta }=\omega _{\beta }}$, and 1 ≤ α < ω_β implies α · ω_β = ω_β, and 2 ≤ α < ω_β implies α^ω_β = ω_β. Using the Veblen hierarchy, β ≠ 0 and α < ω_β imply ${\displaystyle \varphi _{\alpha }(\omega _{\beta })=\omega _{\beta }\,}$ 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.

• Aleph number

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