# 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*: $\S 7.3$, $\S 7.4$