Equivalence of Definitions of Ordinal/Definition 3 is equivalent to Definition 4

Theorem

The following definitions of the concept of Ordinal are equivalent:

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

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Sources

• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $5$: Ordinal Numbers: $\S 2$ Ordinals and transitivity: Exercise $2.3$