Remark 4.7.2.6. Let $\kappa $ be an ordinal. The following conditions are equivalent:
- $(1)$
The ordinal $\kappa $ is a cardinal. That is, there exists a set $S$ such that $\kappa = |S|$.
- $(2)$
For every well-ordered set $(S, \leq )$ of order type $\kappa $, we have $\kappa = |S|$.
- $(3)$
The set of ordinals $\mathrm{Ord}_{< \kappa }$ has cardinality $\kappa $.
See Corollary 4.7.1.23.