Definition:Transfinite Ordinal
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 a 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.
The term transfinite number can also sometimes be seen.
The very word transfinite is a holdover from the tentative steps made by Georg Cantor, seeking a language to communicate in which would minimise the challenge of the preconceived notion of the word infinite.
The word infinite number is used sporadically in $\mathsf{Pr} \infty \mathsf{fWiki}$, which often loosely means a number (of whatever type) which is not finite, and used in a context where the meaning is apparent.
Also see
- Results about transfinite ordinals can be found here.
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Cantor's theory of sets
- 2008: Paul Halmos and Steven Givant: Introduction to Boolean Algebras ... (previous) ... (next): Appendix $\text{A}$: Set Theory: Natural and Ordinal Numbers
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Cantor's theory of sets