Bijection between $\mathbb{N}$ and $[0,\alpha]$

Question

Suppose $\alpha<\omega_1$ is an ordinal. Can anyone give me an example of a bijection between $\mathbb{N}$ and $[0,\alpha]:=\{\gamma: \gamma\leq \alpha\}$. Is there an order preserving bijection between the two sets?

Answer 1

There is a bijection iff $\omega\le\alpha<\omega_1$. There is no order-preserving bijection for any $\alpha$, since $[0,\alpha]$ has a largest element, and $\Bbb N$ does not. Actually producing a bijection between $\Bbb N$ and $[0,\alpha]$ will depend on the specific $\alpha$. For instance, one bijection from $\Bbb N$ to $[0,\omega+\omega]$ sends $0$ to $\omega+\omega$, $2n$ to $n$ if $n>0$, and $2n+1$ to $\omega+n$:

$$f:\Bbb N\to\omega+\omega:n\mapsto\begin{cases} \omega+\omega,&\text{if }n=0\\ k,&\text{if }n=2k>0\\ \omega+k,&\text{if }n=2k+1\;. \end{cases}$$