Definition of a recursive ordinal

Question

I'm having trouble understanding the definition of reclusive or computable ordinal - Wikipedia defines it as follows:

"...an ordinal $\alpha$ is said to be recursive if there is a recursive well-ordering of a subset of the natural numbers having the order type $\alpha$."

I find this hard to picture. Is there a different way of explaining it, or some examples?

Answer 1

Think in "recursive..." as explicit. Example: let be $R$ the relation in $\omega$ defined by $$\forall n>0:\ n\,R\,0,$$ $$\forall m,n>0:\ n\,R\,m\iff n<m.$$ The order type of $(\omega,R)$ is $\omega+1$. Another example: by an explicit bijection $\omega\longrightarrow\omega\times\omega$ you can define in $\omega$ an order of type $\omega^2$.