Different types of well-ordering of $\mathbb{N}$

Question

The question is: "Find 3 non-isomorpic with each other well-orderings of $\mathbb{N}$. Define their ordinals and arrange them in magnitude." The question in ZFC, but I only know that such well-ordering exists due to AC. How do I answer this question?

Answer 1

On $\Bbb{N}$ you can give well-orderings explicitely.

One is the standard order ("natural ordering") on $\Bbb{N}$.

Another would be the standard ordering on $\Bbb{N} \setminus \{1\}$ but with $n < 1$ for all $n \in \mathbb{N} \setminus \{1\}$ (formalize this!)

This new ordering is not isomorphic to the other one, because $\Bbb{N}$ with the natural ordering has no maximal element.

I leave it to you to find a third different ordering.