# Definition:Transfinite Ordinal

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## Contents

## Definition

Let $\alpha$ be an ordinal.

Then $\alpha$ is said to be **transfinite** if and only if it is an infinite set.

### Countable Ordinal

Let $\alpha$ be an ordinal.

Then $\alpha$ is said to be **countable** if and only if it is an countable set.

### Uncountable Ordinal

Let $\alpha$ be an ordinal.

Then $\alpha$ is said to be **uncountable** if and only if it is an uncountable set.

## Also known as

A **transfinite ordinal** is also known as an **infinite ordinal**.

## Also see

## Sources

- 2008: Paul Halmos and Steven Givant:
*Introduction to Boolean Algebras*... (previous) ... (next): Appendix $\text{A}$: Set Theory: Natural and Ordinal Numbers