Equivalence of Definitions of Ordinal
Theorem
The following definitions of the concept of Ordinal are equivalent:
Definition 1
Let $S$ be a set.
Let $\Epsilon \! \restriction_S$ be the restriction of the epsilon relation on $S$.
Then $S$ is an ordinal if and only if:
- $S$ is a transitive set
- $\Epsilon \! \restriction_S$ strictly well-orders $S$.
Definition 2
Let $A$ be a set.
Then $A$ is an ordinal if and only if $A$ is:
Definition 3
An ordinal is a strictly well-ordered set $\struct {S, \prec}$ such that:
- $\forall a \in S: S_a = a$
where $S_a$ is the initial segment of $S$ determined by $a$.
From the definition of an initial segment, and Ordering on Ordinal is Subset Relation, we have that:
- $S_a = \set {x \in S: x \subsetneqq a}$
From Initial Segment of Ordinal is Ordinal we have that $S_a$ is itself an ordinal.
Proof
Definition 1 is equivalent to Definition 2
This follows immediately from the definition of a strict well-ordering.
$\Box$
Definition 1 implies Definition 3
Let $S$ be an ordinal according to Definition 1.
Let $a \in S$.
Then:
\(\ds S_a\) | \(=\) | \(\ds \set {x \in S: x \in_S a}\) | Definition of Initial Segment | |||||||||||
\(\ds \) | \(=\) | \(\ds \set {x: x \in S \land x \in a}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds S \cap a\) | Definition of Set Intersection | |||||||||||
\(\ds \) | \(=\) | \(\ds a\) | as $a \subseteq S$ by transitivity |
$\Box$
Definition 3 implies Definition 1
Let $\struct {S, \prec}$ be an ordinal according to Definition 3.
Let $a \in S$.
Then $a = S_a \subseteq S$ and so $S$ is transitive.
Also, by the definition of set equality:
\(\ds \forall x: \, \) | \(\ds x \in a\) | \(\iff\) | \(\ds x \in S_a\) | |||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds \forall x: \, \) | \(\ds x \in a\) | \(\iff\) | \(\ds \paren {x \in S \land x \prec a}\) |
It has been shown that if $x, a \in S$ then:
- $x \in a \iff x \prec a$
Therefore, $\operatorname \prec = \struct {S, S, R}$ where:
\(\ds R\) | \(=\) | \(\ds \set {\tuple {x, a} \in S \times S: x \prec a}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {\tuple {x, a} \in S \times S: x \in a}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \in_S\) | Definition of Epsilon Restriction |
Hence $\operatorname \prec = \Epsilon {\restriction_S}$.
$\blacksquare$
Sources
- 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 7.3$, $\S 7.4$