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

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## 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 $2.3$