Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

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.