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
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$\blacksquare$
Sources
- 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 7.3$, $\S 7.4$