Reading an article, I see: $$\mathbb{Z}=\mathbb{N}\times\mathbb{N}/\sim\\(a,b)\sim(c,d)\iff a+d=b+c$$
In this formulation of the integers, what would a number actually look like? I’m familiar with the set theory treatment of the naturals, that is $0=\emptyset,1=\{0\},2=\{0,1\},3=\{0,1,2\}\cdots$ but here would the equivalence class $(0,1)\sim(1,2)\sim(2,3)\sim\cdots$ be the equivalence class representing the integer $1$? And would $-1$ be represented by $(1,0)\sim(2,1)\sim\cdots$? And how could one express (I assume with a basis in ZF theory) this article’s formulation of $1$ in terms of purely sets? This ordered pairs notion means what, exactly? $1\in\Bbb Z=\{\{0,1\},\{1,2\},\{2,3\},\cdots\}$? Wouldn’t this contradict the notion $1\in\Bbb N=\{\emptyset\}$?
Additionally, this equivalence relation assumes commutativity of natural arithmetic. Building up from scratch, axiomatically, this article seems to have a flimsy construction of the integers - what is the canonical way?