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Thevon Neumann cardinal assignment is acardinal assignment that usesordinal numbers. For awell-orderable setU, we define itscardinal number to be the smallest ordinal numberequinumerous toU, using the von Neumann definition of an ordinal number. More precisely:

${\displaystyle |U|=\mathrm{c}\mathrm{a}\mathrm{r}\mathrm{d}(U)=inf\{\alpha \in \mathrm{O}\mathrm{N}|\alpha {=}_{c}U\},}$

where ON is theclass of ordinals. This ordinal is also called theinitial ordinal of the cardinal.

That such an ordinal exists and is unique is guaranteed by the fact thatU is well-orderable and that the class of ordinals is well-ordered, using theaxiom of replacement. With the fullaxiom 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.

Each ordinal has an associatedcardinal, its cardinality, obtained by simply forgetting the order. Any well-ordered set having that ordinal as itsorder 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. Theaxiom 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 ordinalis a cardinal.

The${\displaystyle \alpha }$-th infinite initial ordinal is written${\displaystyle {\omega }_{\alpha }}$. Its cardinality is written${\displaystyle {\mathrm{\aleph }}_{\alpha }}$ (the${\displaystyle \alpha }$-thaleph number). For example, the cardinality of${\displaystyle {\omega }_0=\omega }$ is${\displaystyle {\mathrm{\aleph }}_0}$, which is also the cardinality of${\displaystyle {\omega }^2}$,${\displaystyle {\omega }^{\omega }}$, and${\displaystyle {ϵ}_0}$ (all arecountable ordinals). So we identify${\displaystyle {\omega }_{\alpha }}$ with${\displaystyle {\mathrm{\aleph }}_{\alpha }}$, except that the notation${\displaystyle {\mathrm{\aleph }}_{\alpha }}$ is used for writing cardinals, and${\displaystyle {\omega }_{\alpha }}$ for writing ordinals. This is important becausearithmetic on cardinals is different fromarithmetic on ordinals, for example${\displaystyle {\mathrm{\aleph }}_{\alpha }^2}$ = ${\displaystyle {\mathrm{\aleph }}_{\alpha }}$ whereas${\displaystyle {\omega }_{\alpha }^2}$ > ${\displaystyle {\omega }_{\alpha }}$. Also,${\displaystyle {\omega }_1}$ is the smallestuncountable ordinal (to see that it exists, consider the set ofequivalence 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${\displaystyle {\mathrm{\aleph }}_1}$, 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, implies${\displaystyle \alpha +{\omega }_{\beta }={\omega }_{\beta }}$, and 1 ≤α < ω_β impliesα · ω_β = ω_β, and 2 ≤α < ω_β impliesα^ω_β = ω_β. Using theVeblen hierarchy,β ≠ 0 andα < ω_β imply${\displaystyle {\phi }_{\alpha }({\omega }_{\beta })={\omega }_{\beta }}$ 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. MoschovakisNotes on Set Theory (1994 Springer) p. 198

• Cardinal numbers
• Ordinal numbers

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