Cardinal Number

DEF-CA. Cardinal Number.

• An ordinal $\alpha$ is a cardinal number / cardinal if

  \[\forall\beta<\alpha:\beta\lnequ\alpha.\]

• The class of all cardinals is denoted by $\Card$.

• Every natural number is a cardinal and is called a finite cardinal.

• A cardinal is infinite if it is not finite.

PROP-CA-P.

$\Card$ is a proper class.

PROP-CA-LO.

Every infinite cardinal is a limit ordinal.

Proof.By PROP-SUC-CARD.

PROP-CA-G.

For every ordinal $\alpha$ there is a cardinal greater than $\alpha$.