TITLE: how to create the set of hyperreal numbers using ultraproduct
QUESTION [5 upvotes]: As title says, can anyone explain how to create the set of hyperreal numbers using ultraproduct from the set of real umbers?

REPLY [5 votes]: The explanation may be a bit simpler than seems to be implied by earlier answer. Namely, the hyperreals are a certain quotient of the space $\mathbb{R}^{\mathbb N}$ of sequences of real numbers.  More specifically, consider sequences $(x_n)$ that vanish for almost all $n$, and let $I\subset \mathbb{R}^{\mathbb N}$ be a maximal ideal containing all such sequences (the existence of a maximal ideal is proved in undergraduate algebra courses).  Then the hyperreals are a quotient field $\mathbb{R}^{\mathbb N}/I$.  This is equivalent to a certain construction using an ultrafilter on $\mathbb N$. A more detailed summary of the construction can be found on page 911 in  https://www.math.wisc.edu/~keisler/calc.html