What does this notation mean in sets?

Question

If I have the following set $E_n$ where n denotes a natural number. What does putting it in subscript mean ?

Thanks

Edit :

Difference of two natural integers:

For all integers $m,n \in \mathbb N$ which satifies $m>n$, we have $m-n \in\mathbb N$. Indeed, for all $n \in \mathbb N$, the set $$E_n=\{m-n \in \mathbb N : m \in \mathbb N, m>n\}$$ contains ($n+1)-n = 1+ (n-n) = 1+ 0 = 1$ with $n+1>n+0 = n$. Furthermore, if $k$ is element of $E_n$, there exists $m \in \mathbb N$ with $m>n$ such that $$k+1= (m-n)+1= (m+1)-n$$ with $m+1 \in \mathbb N$ satisfying $m+1>n+1>n+0=n$ where $k+1 \in E_n$

Answer 1

Intuitively, it means that $E_0$, $E_1$, $E_2$ and so forth are names of different sets -- though presumably ones that you're going to say something general about later on.

Formally, it is just a different notation for saying you have some function $E:\mathbb N\to\text{something}$, where you're declaring that by convention the function values written $E_n$ rather than $E(n)$.