Condition for Set Equivalent to Cardinal Number
Theorem
Let $S$ be a set.
Let $\card S$ denote the cardinality of $S$.
That is, let $\card S$ be the intersection of all ordinals equivalent to $S$.
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Note that in the absence of the Axiom of Choice, $\card S$ may be the class of all sets.
Then the following are equivalent:
- $(1): \quad S \sim \card S$
- $(2): \quad \exists x \in \On: S \sim x$
- $(3): \quad \exists x \in \On: \exists y: \paren {y \subseteq x \land S \sim y}$
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 $\exists x \in \On: S \sim x$, then by Class of All Ordinals is Well-Ordered by Subset Relation 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 $z \in \On$ by Unique Isomorphism between Ordinal Subset and Unique Ordinal.
Therefore, by Set Equivalence behaves like 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 $x$.
\(\ds \bigcap \set {x \in \On : S \sim x}\) | \(=\) | \(\ds \bigcap \O\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \mathbb U\) | Intersection of Empty Set |
But $S \not \sim \mathbb U$, so $S \not \sim \card S$ 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: $\S 10.9$