# Condition for Set Equivalent to Cardinal Number

## Theorem

Let $S$ be a set.

Let $\left\vert{S}\right\vert$ denote the cardinal number of $S$.

That is, let $\left\vert{S}\right\vert$ be the intersection of all ordinals equivalent to $S$.

Note that in the absence of the Axiom of Choice, $\left\vert{S}\right\vert$ may be the class of all sets.

Then the following are equivalent:

- $(1): \quad S \sim \left\vert{S}\right\vert$
- $(2): \quad \exists x \in \operatorname{On}: S \sim x$
- $(3): \quad \exists x \in \operatorname{On}: \exists y: \left({y \subseteq x \land S \sim y}\right)$

where $A \sim B$ means that there is a bijection from $A$ onto $B$, and the quantification over $y$ is unbounded (so $y$ may be any set).

## Proof

### $2 \implies 1$

If $\exists x \in \operatorname{On}: S \sim x$, then by Ordinal Class is Strongly Well-Ordered by Subset there is a smallest ordinal $x$ such that $S \sim x$.

This smallest ordinal $x$ is the cardinal number of $S$, by definition.

$\Box$

### $3 \implies 2$

Suppose that $y \subseteq x$ and $S \sim y$ for some ordinal $x$.

Since $y \subseteq x$, it follows that $y \sim z$ for some $z \in \operatorname{On}$ by Unique Isomorphism between Ordinal Subset and Unique Ordinal.

Therefore, by Set Equivalence is Equivalence Relation, $S \sim z$.

$\Box$

### $1 \implies 3$

Suppose that $3$ is not true.

It follows that $S \not \sim x$ for any ordinal $x$.

\(\displaystyle \bigcap \left\{ {x \in \operatorname{On} : S \sim x}\right\}\) | \(=\) | \(\displaystyle \bigcap \varnothing\) | $\quad$ | $\quad$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \mathbb U\) | $\quad$ by Intersection of Empty Set | $\quad$ |

But $S \not \sim \mathbb U$, so $S \not \sim \left\vert{S}\right\vert$ by the definition of cardinal number.

The result follows by contraposition.

$\blacksquare$

## Sources

- 1971: Gaisi Takeuti and Wilson M. Zaring:
*Introduction to Axiomatic Set Theory*: $\S 10.9$