Ultrapower construction of ${}^{\ast \ast}\Bbb R$

Question

Suppose, we want to add positive elements that are smaller than all positive elements in ${}^{\ast}{\Bbb R}$. One way to do this, as shown in this master’s thesis, is to construct sequences of elements in ${}^{\ast}{\Bbb R}$ that range over ${}^{\ast}{\Bbb N}$ and find an ultrafilter on ${}^{\ast}{\Bbb N}$, ${}^{\ast}{\mathcal U}$.

Can we find such ultrafilter ${}^{\ast}{\mathcal U}$? Do we need to require all sets in ${}^{\ast}{\mathcal U}$ to be internal sets? Or we have to live with a cheap version, ${}^{\ast}\mathcal {P}{(\Bbb N)} \cap {}^{\ast}{\mathcal U}$?

The author want to eliminate the possibility of constructing an positive element in ${}^{\ast}\Bbb R$ that is strictly smaller than all positive elements in ${}^{\ast\ast}\Bbb R$ by excluding $\Bbb N$ from ${}^{\ast} \mathcal U$. But it seems to me it's far from enough. ${}^{\ast} \mathcal U$ shouldn't contain any element with the cardinality of $\Bbb N$. I think what he need is a uniform ultrafilter in which all elements have the cardinality of $2^{\aleph_0}$. Is it right?

Answer 1

$^\star\mathbb{R}$ only satisfies countable saturation, so to defeat it it suffices to take an index set of cardinality the continuum in the ultraproduct construction, so using $^\star\mathbb{N}$ is fine. It is not entirely clear to me why one needs such a $^{\star\star}\mathbb{R}$.

Answer 2

Why would you need such a thing as **R? To define a *Riemann integral over infinitesimal intervals. And why would one want to do that? Because that is one way to integrate over fractals, where, for instance, the Cantor set is 2^omega intervals of length 2^-omega. Now integrate your function over each of these intervals, and add them all up.