Describe all non negative integers using only the constant symbols $0,1$, operations $+,\cdot$ and relation $=$ on $\mathbb Z$.
In the reals, a similar description is simple, we just say $x\geq 0\iff \exists y: y\cdot y = x$.
In the integers, however, that doesn't work. There's no $\sqrt{2}\in\mathbb Z$, for instance.
We could say $$x\geq 0 \iff x=0\lor x=1\lor x=1+1\lor\ldots $$ but that's not satisfactory since it's not finite.
I always seem to be stuck defining it via itself, provided the expression is finite. What am I missing?
It would also be sufficient to describe negative integers.
As per orangeskid's reference. Lagrange's theorem does the trick.