Succinct notation for the set of all initial segments of $\mathbb{N}$?

Question

The $j^{th}$ initial segment of $\mathbb{N}$, i.e. $\{i:i\leq j\}$ is typically denoted by $\mathbb{N}_{\leq j}$. One can easily denote the set of all such segments by $\{\mathbb{N}_{\leq j}, j\in\mathbb{N}\}$. Is there a succincter "standard" way, something like $\mathbb{N}_{\leq\mathbb{N}}$?

Answer 1

There is no standard notation for this.

In the von Neumann construction of natural numbers, $n$ is the set $\{0,1,\dots, n-1\}$. This is clever and elegant but likely to be confusing for readers, unless your topic is exactly the set-theoretical construction of numbers.

Better be clear than use non-standard notation, however clever.