Definition:Transfinite Ordinal

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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