A partial order on $\mathbb{N}$ that is not total

Question

In the text Modern Real Analysis by Ziemer there is a question that asks to use the "natural partial order" on $\mathcal{P}(\{1,2,3\})$ to obtain a partial order on $\mathbb{N}$.

I have scratched my head for some time but have failed to find a link between $\subset$ which is supposedly the natural partial order on $\mathcal{P}(\{1,2,3\})$ and a possible partial order on $\mathbb{N}$.

EDIT: I must mention that the question asks to find a partial order that is not total.

Answer 1

You can say that, given $a,b\in\mathbb{N}$, $a$ precedes $b$ if $b$ is divisible by $a$. This is a partial order, but not a total order because for example $6$ doesn't divide $7$ and vice versa.