What is the maximal ideal used to construct the Hyperreals?

Question

I'm working through the AMM paper here on the construction of (a model of) the Hyperreals. The paper gives the following sequence of arguments and definitions:

If we take the set of sequences $\mathbb{R}^\mathbb{N}$, define $F$ as the set of sequences with finite support on $\mathbb{N}$ (the ones that have identically zero tails). We know there must be a maximal ideal $M$ that contains $F$ since $F$ is a proper ideal. We can then define a field $\mathbb{R}^* = \mathbb{R}^\mathbb{N} / M$ by taking the quotient of the set of sequences by that maximal ideal, and we call $\mathbb{R}^*$ the Hyperreals.

Now I'm with you, that all makes sense (admittedly I didn't know every proper ideal is contained in a maximal ideal, but it makes sense). However I'm wondering what the hell $M$ actually looks like? What are its elements?

Answer 1

Too long for a comment:

1. First off, lisyarus' comment is spot-on, that in general it can be hard to get explicit descriptions of things constructed via the axiom of choice. However, in this particular case, there are several things that can be said...
2. At a very primitive level, the construction of the hyperreals is very much like the construction of the reals from rational numbers via Cauchy sequences. Each point of $\mathbb{R}^*$, as defined as in your question, is an equivalence class of sequences, where two sequences are equivalent if e.g. they are identical after some finite number of terms. By comparison, in the construction of the real numbers from rationals, two Cauchy sequences of rationals are equivalent if they get arbitrarily close and stay close for high indices (the usual Cauchy criterion).
3. A better way to get a feel for what $\mathbb{R}^*$ looks like is from carrying on the theory of nonstandard analysis, which leads to the discovery that $\mathbb{R}^*$ looks roughly like the real line, but augmented with clouds of infinitesimals around each point, and infinitely large numbers too.

As an aside, it is often true that non-standard analysis gives us a tool to understand abstract objects created by the axiom of choice. I can't find a reference to this right now, but I know there are instances of standard constructions (i.e. not using the hyperreals) which need the axiom of choice and are somewhat hard to understand, but which can be constructed explicitly and "constructively" within the framework of non-standard analysis (i.e. using infinitesimal and infinite hyperreals). I put "constructively" in quotes because the hyperreals require the axiom of choice for their definition, so such "constructions" are still not properly constructive, but once you've allowed yourself the possibility of non-standard numbers, then you can sometimes use them to get intuition about otherwise vague objects.