Cardinal Number Equivalence or Equal to Universe

It has been suggested that this page be renamed. In particular: Not sure what to, but current name is clumsy and difficult to understand To discuss this page in more detail, feel free to use the talk page.

Theorem

Let $S$ be a set.

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

Let $\mathbb U$ denote the universal class.

Then:

$S \sim \card S \lor \card S = \mathbb U$

Proof

By Condition for Set Equivalent to Cardinal Number:

If $\exists x \in \On: S \sim x$, then:

$S \sim \card S$

If $\neg \exists x \in \On: S \sim x$, then:

\(\ds \bigcap \set {x \in \On : S \sim x}\)

\(=\)

\(\ds \bigcap \O\)

\(\ds \)

\(=\)

\(\ds \mathbb U\)

Intersection of Empty Set

And thus by the definition of cardinal number:

$\card S = \mathbb U$

$\blacksquare$

Sources

• 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 10.10$