Definition 1. An ordinal $\alpha$ is a cardinal number or cardinal if
\[\forall \beta < \alpha : \beta \lnequ \alpha.\]The class of all cardinals is denoted by $\Card$.
Definition 2. Every natural number is a cardinal, called a finite cardinal. A cardinal is infinite if it is not finite.
Proposition 1. $\Card$ is a proper class.
Proof
Show that $\Ord \subseteq \bigcup \Card$.
Proposition 2. Every infinite cardinal is a limit ordinal.
References
- [1]
- Wikipedia contributors. Cardinal number. Wikipedia, The Free Encyclopedia.
- [2]
- Thomas Jech. Set Theory. 3rd Millennium ed, rev. and expanded. Springer, 2002. p. 29.