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Indexing classes of ordinals

Any well-ordered set is similar (order-isomorphic) to a unique ordinal number α {\displaystyle \alpha }; in other words, its elements can be indexed in increasing fashion by the ordinals less than α {\displaystyle \alpha }. This applies, in particular, to any set of ordinals: any set of ordinals is naturally indexed by the ordinals less than some α {\displaystyle \alpha }. The same holds, with a slight modification, for classes of ordinals (a collection of ordinals, possibly too large to form a set, defined by some property): any class of ordinals can be indexed by ordinals (and, when the class is unbounded in the class of all ordinals, this puts it in class-bijection with the class of all ordinals). So the γ {\displaystyle \gamma } -th element in the class (with the convention that the "0-th" is the smallest, the "1-st" is the next smallest, and so on) can be freely spoken of. Formally, the definition is by transfinite induction: the γ {\displaystyle \gamma } -th element of the class is defined (provided it has already been defined for all β < γ {\displaystyle \beta <\gamma }), as the smallest element greater than the β {\displaystyle \beta } -th element for all β < γ {\displaystyle \beta <\gamma }.

This could be applied, for example, to the class of limit ordinals: the γ {\displaystyle \gamma } -th ordinal, which is either a limit or zero is ω ⋅ γ {\displaystyle \omega \cdot \gamma } (see ordinal arithmetic for the definition of multiplication of ordinals). Similarly, one can consider additively indecomposable ordinals (meaning a nonzero ordinal that is not the sum of two strictly smaller ordinals): the γ {\displaystyle \gamma } -th additively indecomposable ordinal is indexed as ω γ {\displaystyle \omega ^{\gamma }}. The technique of indexing classes of ordinals is often useful in the context of fixed points: for example, the γ {\displaystyle \gamma } -th ordinal α {\displaystyle \alpha } such that ω α = α {\displaystyle \omega ^{\alpha }=\alpha } is written ε γ {\displaystyle \varepsilon _{\gamma }}. These are called the " epsilon numbers ".