Ordinal Number Equivalent to Cardinal Number

Theorem

Let $x$ be an ordinal.

Let $\card x$ denote the cardinal number of $x$.

Then:

$x \sim \card x$

where $\sim$ denotes set equivalence.

Proof

From Set is Equivalent to Itself:

$x \sim x$

Therefore, $x$ is equivalent to some ordinal.

By Condition for Set Equivalent to Cardinal Number:

$x \sim \card x$

$\blacksquare$

Sources

• 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 10.11$