Equivalence of Definitions of Ordinal

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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


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$\blacksquare$


Sources

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