Equivalence of Definitions of Ordinal

Theorem

The following definitions of the concept of Ordinal are equivalent:

Definition 1

$\alpha$ is an ordinal if and only if it fulfils the following conditions:

 $(1)$ $:$ $\alpha$ is a transitive set $(2)$ $:$ $\Epsilon {\restriction_\alpha}$ strictly well-orders $\alpha$

where $\Epsilon {\restriction_\alpha}$ is the restriction of the epsilon relation to $\alpha$.

Definition 2

$\alpha$ is an ordinal if and only if it fulfils the following conditions:

 $(1)$ $:$ $\alpha$ is a transitive set $(2)$ $:$ the epsilon relation is connected on $\alpha$: $\ds \forall x, y \in \alpha: x \ne y \implies x \in y \lor y \in x$ $(3)$ $:$ $\alpha$ is well-founded.

Definition 3

An ordinal is a strictly well-ordered set $\struct {\alpha, \prec}$ such that:

$\forall \beta \in \alpha: \alpha_\beta = \beta$

where $\alpha_\beta$ is the initial segment of $\alpha$ determined by $\beta$:

$\alpha_\beta = \set {x \in \alpha: x \prec \beta}$

Definition 4

$\alpha$ is an ordinal if and only if:

$\alpha$ is an element of every superinductive class.

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 $\alpha$ be an ordinal according to Definition 1.

Let $\beta \in \alpha$.

Then:

 $\ds \alpha_\beta$ $=$ $\ds \set {x \in \alpha: x \in_\alpha \beta}$ Definition of Initial Segment $\ds$ $=$ $\ds \set {x: x \in \alpha \land x \in \beta}$ $\ds$ $=$ $\ds \alpha \cap \beta$ Definition of Set Intersection $\ds$ $=$ $\ds \beta$ as $\beta \subseteq \alpha$ by transitivity

$\Box$

Definition 3 implies Definition 1

Let $\struct {\alpha, \prec}$ be an ordinal according to Definition 3.

Let $\beta \in \alpha$.

Then $\beta = \alpha_\beta \subseteq \alpha$ and so $\alpha$ is transitive.

Also, by the definition of set equality:

 $\ds \forall x: \,$ $\ds x \in \beta$ $\iff$ $\ds x \in \alpha_\beta$ $\ds \leadstoandfrom \ \$ $\ds \forall x: \,$ $\ds x \in \beta$ $\iff$ $\ds \paren {x \in \alpha \land x \prec \beta}$

It has been shown that if $x, \beta \in \alpha$ then:

$x \in \beta \iff x \prec \beta$

Therefore, ${\prec} = \struct {\alpha, \alpha, \RR}$ where:

 $\ds \RR$ $=$ $\ds \set {\tuple {x, \beta} \in \alpha \times \alpha: x \prec \beta}$ $\ds$ $=$ $\ds \set {\tuple {x, \beta} \in \alpha \times \alpha: x \in \beta}$ $\ds$ $=$ $\ds \in_\alpha$ Definition of Epsilon Restriction

Hence ${\prec} = \Epsilon {\restriction_\alpha}$.

$\Box$

Definition 3 is equivalent to Definition 4

$\blacksquare$