How big is the set of hyper-naturals? Consider the set $\mathbb N^*$, the set of hypernaturals. How big is this set? Is it the same size as $\mathbb R^*$?
 A: 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 $|\Bbb N^*|=|\Bbb R|$.
A: Here is a simple proof that $|ℕ^*| ≥ |ℝ|$, without relying on the construction of the hyper-reals.
Let $H$ be an infinite hyper-natural.
Let $r, r'$ be non-negative real numbers where $r ≠ r'$. Then $|r - r'| > 0$ is not infinitesimal. Thus, $H|r - r'|$ is infinite and in particular, $|Hr - Hr'| > 1$. Therefore, $\lfloor Hr\rfloor ≠ \lfloor Hr' \rfloor$. So, the following function is one-to-one:
$$f : ℝ_{≥0} → ℕ^* f(x) = \lfloor Hx\rfloor$$ 
