BI: nonstandard analysis, a small investment

First of all, you can read about nonstandard analysis in a slightly more famous blog than my own, for example you can check out this post. I guess I should explain the quote “a small investment” by Isaac Goldbring (afaik). Using nonstandard analysis is never strictly necessary. However, it provides such a clearer view on a broad range of constructions that it is a valuable tool, or at the very least a valuable aid to intuition. One of the advantages of nonstandard analysis is that it gives a way of formalizing intuitive concepts involving infinities that are otherwise rather clumsy to define formally. A very nice example I’ve recently learned about is given by ends of groups (and other metric spaces). Those are “connected components at infinity”, and with nonstandard analysis you can literally define them this way, while standard definitions are more obscure, in my opinion at least. And then my favourite: asymptotic cones… I always think of asymptotic cones from the nonstandard viewpoint, while most often I (painfully) translate my thoughts into the language of ultrafilters&co when writing a paper.
The adjective “small” is referred to the FACT that it really doesn’t take much to learn the basics of nonstandard analysis, and so here we go!

Let’s start with $\mathbb{R}$. The main idea is to extend it to some object $^*\mathbb{R}\supseteq \mathbb{R}$, its nonstandard extension, that contains infinitesimal and infinite quantities. Admit it: your intuition on limits is at least partly based on them!
$^*\mathbb{R}$ should also look as similar as possible to $\mathbb{R}$. More in general, one might look for an object $^*X$ associated to, say, a metric space $X$ that encodes the behaviour of $X$ at very large and very small scales. As it turns out, you can construct $^*X\supseteq X$ starting from any set $X$. Also, you can take nonstandard extensions of functions $f:X\to Y\rightsquigarrow \,^*f:\,^*X\to\,^*Y$ and relations (orders, etc.), so that whenever you have a structure on a set $X$ (group, metric space,…) you get something for $^*X$, we’ll see exactly what. I’ll call standard world (a large enough bit of) the usual universe of set theory and nonstandard world the union of all nonstandard extensions of stuff in the standard world. As it turns out, standard and nonstandard world are spectacularly similar.

Slogan: You can’t tell apart the standard world from the nonstandard world just living inside them.

(The slogan translates into model theoretic language, modulo technicalities, as “the natural map from the standard world to the nonstandard world is an elementary embedding”, so in particular “$X$ and $^*X$ satisfy the same first order properties”.) In order to explain the slogan, we have to introduce the concept of internal set. Bear with me, this is the non-intuitive part of the business (“small investment”, not “gift”). Internal sets are the subsets of $^*X$ that you can see when living inside it. Formally, those are the sets in $^*\mathcal{P}(X)$, but that’s not very enlightening. More interestingly:
1) nonstandard extensions of subsets of $X$ are internal subsets of $^*X$,
2) finite sets are internal,
3) you can show that a set is internal if you show that it satisfies a property defined in terms of other internal sets.
There are also have internal functions and relations, that satisfy similar properties.

Examples time. Intervals in $^*\mathbb{R}$, say closed intervals, are internal. (The nonstandard extension of the total order of $\mathbb{R}$ is an total order on $^*\mathbb{R}$ as we will see in a minute, so “closed interval” makes sense. Also, it is the nonstandard extension of something, so it is internal.) Well, a closed interval is defined in terms of the order structure of $^*\mathbb{R}$ by the property “the set of all elements lying between the endpoints”. Other examples of internal sets are given by level sets of internal functions… Those are defined in terms of other internal objects, aren’t they?

The reason why internal sets (and functions and relations) are interesting is the following theorem by Łoś:

Transfer principle: Internal sets, functions and relations satisfy the same formulae that are satisfied in the standard world.

Here are some examples to clarify this. Let’s start with the simplest type of internal guys, the nonstandard extensions of guys in the standard world. A group structure on a set is an operation satisfying  certain properties. Now, if you take the nonstandard extension of the group operation, by the transfer principle it will also satisfy those properties. So, the nonstandard extension of a group is a group. Similarly, the nonstandard extension of a (total) order is a (total) order.
Now, a slightly more sophisticated example. Every non-empty subset of $\mathbb{N}$ has a minimum. Hence, every internal non-empty subset of $\mathbb{N}$ has a minimum. There are non-internal non-empty subsets of $\mathbb{N}$ with no minimum, but you cannot see them if you live in the nonstandard world as they are not internal. And this explains the slogan.

So far we have explored the similarities between the standard and the nonstandard world, but of course we would like the nonstandard world to have extra properties. For example, we would like $^*\mathbb{R}$ to contain infinitesimals.
The feature of the nonstandard world that makes this true is called saturation:

Saturation: the intersection of countably many internal sets $\{A_i\}$ is non-empty as long as each intersection $A_0\cap \dots \cap A_n$ is non-empty.

A good way of thinking about this is: if I have countably many conditions so that every finite collection of them can be satisfied, then all of them can be satisfied.
The existence of infinitesimals follows applying saturation to the internal sets $(0,1/i)$.

I would like to conclude with an example where these concepts are applied, except that the authors didn’t use the language of nonstandard analysis directly. When $G$ is a group, we say that $G$ satisfies the law $w$, where $w$ is a non-trivial word in some alphabet $a_1,\dots, a_n$, if whenever the letters in $w$ are substituted by elements of $G$ one obtains the identity. For example, abelian groups satisfy the law $a_1a_2a_1^{-1}a_2^{-1}$, and similarly any solvable group satisfies a law as well.

[Lemma 6.15, Druţu-Sapir] A group $G$ satisfies a law if and only if $^*G$ does not contain a free group on two generators.

The only if part follows applying transfer: if $G$ satisfies a law then $^*G$ satisfies the same law. Once again: one cannot distinguish $G$ from $^*G$ using internal concepts, and satisfying a given law is indeed an internal concept because in the appropriate sense any given word is built up in finitely many steps (or, if you prefer, contains a finite amount of information).
The converse relies on saturation. There are countably many words, and not satisfying a law means that you can find for each finite collection of words in two variables $w_1,\dots, w_n$ two elements of your group that do not satisfy any of them. There’s a trick here: you should choose two elements not satisfying the law $[w_1,[w_2,[\dots[w_{n-1}, w_n]\dots]$ (that you can assume to be non-trivial). So, you may not find two elements in $G$ that do not satisfy any law, but by saturation you know that you can find them in $^*G$. And those freely generate a free-group.

Final remarks for those of you who know about ultrafilters. The proofs of the lemma above using ultrafilters and using nonstandard analysis are the same, but somehow the key concept used is saturation, which fits more naturally in the nonstandard setting… Also, if you use the ultrafilters language you have to re-prove saturation every time you use it…