Cardinal of Cardinal Equal to Cardinal/Corollary

Theorem

Let $\NN$ denote the class of all cardinal numbers.

Let $x$ be an ordinal.

Then:

$x \in \NN \iff x = \card x$

Proof

Necessary Condition

Suppose $x = \card x$.

Then $x = \card y$ for some $y$ by Existential Generalisation.

By definition of class of all cardinals:

$\NN = \set {x \in \On: \exists y: x = \card y}$

It follows that $x \in \NN$.

$\Box$

Sufficient Condition

\(\ds x\)

\(\in\)

\(\ds \NN\)

by hypothesis

\(\ds \leadsto \ \ \)

\(\ds x\)

\(=\)

\(\ds \card y\)

Definition of Class of All Cardinals

\(\ds \leadsto \ \ \)

\(\ds \card x\)

\(=\)

\(\ds \card y\)

Cardinal of Cardinal Equal to Cardinal

\(\ds \leadsto \ \ \)

\(\ds x\)

\(=\)

\(\ds \card x\)

Equality is Transitive

$\blacksquare$

Sources

• 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 10.38$