Category:Equivalence of Definitions of Ordinal

This category contains pages concerning Equivalence of Definitions of Ordinal:

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.