How big is the set of hyper-naturals?

Question

Consider the set $\mathbb N^*$, the set of hypernaturals. How big is this set? Is it the same size as $\mathbb R^*$?

Answer 1

This answer to an earlier question shows that $|\Bbb R^*|=|\Bbb R|$. Clearly $|\Bbb N^*|\le|\Bbb R^*|$, so we need only show that $|\Bbb R|\le|\Bbb N^*|$ to complete a proof that $|\Bbb N^*|=|\Bbb R|$.

For $0\le x\in\Bbb R$ let $$\sigma_x=\left\langle\left\lfloor 10^kx\right\rfloor:k\in\Bbb N\right\rangle\in{}^{\Bbb N}\Bbb N\;,$$

and let $\mathscr{U}$ be any free ultrafilter on $\Bbb N$. If $x,y\in\Bbb R_{\ge 0}$ and $x\ne y$, there is an $m\in\Bbb N$ such that $$\left\lfloor 10^kx\right\rfloor\ne\left\lfloor 10^ky\right\rfloor$$ for all $k\ge m$. Thus, if $\mathscr{U}$ is any free ultrafilter on $\Bbb N$, $[\sigma_x]_\mathscr{U}\ne[\sigma_y]_\mathscr{U}$, and the map

$$\Bbb R_{\ge 0}\to\Bbb N^*:x\mapsto[\sigma_x]_\mathscr{U}$$

is injective. It follows immediately that $|\Bbb R|\le|\Bbb N^*|$ and hence that $|\Bbb N^*|=|\Bbb R|$.