2-well-orderings on the same set

Question

Can you find a set $X$, and two well-orderings $<_{1}, <_{2}$ of $X$ such that $(X, <_{1})$ is not isomorphic to $(X,<_{2})$.

I know $X$ cannot be finite because if it were finite then any two well-orderings on X would be isomorphic to each other.( chains with least and greatest elements)

Answer 1

Let $X=\Bbb{N}=\{1,2,3,4,\cdots\}$ and $<_1$ be a usual ordering of the natural number. Let define $<_2$ as follows:

1. $n<_2 1$ for all $n=2,3,4,\cdots$
2. If $m,n\neq 1$ then $m<_2 n$ iff $m<n$.

Then $<_2$ is a well-ordering of $\Bbb{N}$, and it is distinct to the ordering $<_1$, because $<_2$ gives the maximal element, but $<_1$ is not. In fact, there are $\aleph_1$ many distinct well-orderings of $\Bbb{N}$ (up to order isomorphism).