What sets are there with order type $\omega^2$

Question

I’ve recently been thinking about order types of sets, and I can’t come up with a set of natural numbers order type $\omega^2$ or $\omega\cdot\omega$. If someone could help me by giving an example it would be appreciated. -Thanks

Answer 1

How about something like:

$\begin{array}{ccccccccc} 0, & 1, & 3, & 6, & 10, & \dots\\ 2, & 4, & 7, & 11, & 16, & \dots\\ 5, & 8, & 12, & 17, & 23, & \dots\\ 9, & 13, & 18, & 24, & 31, & \dots\\ 14, & 19, & 25, & 32, & 49, & \dots\\ \vdots & \vdots & \vdots & \vdots & \vdots & \ddots\\ \\ \\ \end{array}$

So the set of natural numbers ordered by: $0,1,3,6,10,\dots,2,4,7,11,16,\dots$