Transfinite Numbers

Overview
A transfinite number is a Set that contains an unending number of elements. According to Georg Cantor, there are two classifications of transfinite numbers, transfinite cardinals and transfinite ordinals.

Transfinite Cardinal Numbers
Transfinite cardinal numbers are infinite sets whose elements are proper quantities.

The symbol presented here represents aleph null, the smallest transfinite cardinal. The defining characteristic of the sets nominated by aleph null is their One-to-One Correspondence with ω, the smallest transfinite ordinal. Some examples of aleph null are the set of integers, the set of prime numbers, and the set of even numbers.

The next smallest transfinite cardinal number is aleph one, which is strictly larger than aleph null because Cantor's Diagonal Argument demonstrates that it will always contain elements that exceed the ordinality ω.

There exist infinitely many transfinite cardinal numbers.

Transfinite Ordinal Numbers
Transfinite ordinal numbers are infinite sets who elements are order types of transfinite cardinal numbers.

ω represents the smallest transfinite ordinal, followed by ω1, ω2, and so on. The largest transfinite ordinal is denoted by Ω. It is the first transfinite ordinal, and Cantor called it Absolute Infinity.