Introduction to Aleph Null

I was waiting forever for the lecture on Aleph Null

Bijection between the natural numbers and a set means that the set is countable

Aleph Null
Aleph Null is the cardinality of a countably infinite set

Let 𝓒 be any nonempty collection of sets. Then serts being equivalent is an equivalence relation on 𝒫

Equivalence class
We can represent each equivalence class (for finite sets) through the whole numbers (natural numbers plus zero)

Bijection between all countable sets (obvious?)
Since every countably infinite set has a bijection with the natural numbers, it follows that when using composition every countably infinite set is bijective with every other countably infinite set

Cardinality of real numbers
Cardinality of real numbers is often denoted 𝕮

It is not undefined, and there are things with this cardinality

Cantor's theorem
Suppose