Beth Number - Generalization

Generalization

The more general symbol, for ordinals α and cardinals κ, is occasionally used. It is defined by:

if λ is a limit ordinal.

So

In ZF, for any cardinals κ and μ, there is an ordinal α such that:

And in ZF, for any cardinal κ and ordinals α and β:

Consequently, in Zermelo–Fraenkel set theory absent ur-elements with or without the axiom of choice, for any cardinals κ and μ, the equality

holds for all sufficiently large ordinals β (that is, there is an ordinal α such that the equality holds for every ordinal β ≥ α).

This also holds in Zermelo–Fraenkel set theory with ur-elements with or without the axiom of choice provided the ur-elements form a set which is equinumerous with a pure set (a set whose transitive closure contains no ur-elements). If the axiom of choice holds, then any set of ur-elements is equinumerous with a pure set.