how to create the set of hyperreal numbers using ultraproduct

Question

As title says, can anyone explain how to create the set of hyperreal numbers using ultraproduct from the set of real umbers?

Answer 1

I’d say that a full explanation is beyond the scope of an answer here. Section $3$ of this PDF is a fairly gentle but very incomplete start; this PDF goes into much more detail and seems to be about as accessible as a thorough treatment is likely to be, though I’ve only skimmed it.

Answer 2

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 $\mathbb N$.