This article has more of an analysis and set theory flavour than the last one. I’m trying to keep it fairly light, but it neccesarily creeps in. This is a common theme in this subject – it bounces back and forth between analysis, set theory and algebra. This is one of the reasons I like it.

The hyperreals, R*, were originally studied for model theoretic reasons. They’re a particular example of what’s called an ultrapower, which have the nice property that they’re elementarily equivalent to the original structure (that is to say they satisfy precisely the same first order statements). Note that ‘x is an infinitesimal’ is not first order, because it requires you to quantify over the naturals. Because of this they provide an alternate approach to real analysis and calculus which some people quite like, as it allows one to make sense of a lot of formal manipulations with infinitesimal and infinite quantities. A good example of this is Keisler’s “Elementary Calculus: An Approach Using Infinitesimals”.

I’m not going to go into this in depth here. This is just for background. I’m quite interested in learning more about ultrapowers, so I may at some later date write an article on them as I do so, but it’s really tangential to this subject.

Note: I’ve performed a slight dodge in talking about ‘the’ hyperreals. They actually need not be unique up to isomorphism and can depend on the choice of ultrafilter. This depends on the underlying set theory. I really don’t want to have to worry about this at the moment, so I’m instead going to assume they aren’t unique and refer to them as ultrapowers instead. Later on once we’ve developed a lot of theory which doesn’t depend on the underlying set theory (at least past ZFC. I don’t intend to quibble about the axiom of choice at all) we’ll be able to give an easy condition which guarantees uniqueness.

Alright, enough logic and set theory. I’ve probably scared off all the algebraists. :)

The reals are in some sense quite small. They have a countable dense subset, and they inherit a lot of behaviour from this. In fact, every subset of R has a countable dense subset.

Ultrapowers are on the other hand a lot larger. We have the following result:

Proposition 1:

Let K be an ultrapower and A, B be countable subsets of K such that for every a in A and b in B we have a < b. Then there exists x such that for every a in A and b in B we have a < x < b.
I'm temporarily blanking on the proof of this. It should just be a clever diagonalisation argument. I'll edit it in later.
We'll need to consider such things quite a lot, so we'll introduce the following notation: A << B if for every a in A, b in B we have a < b. We will usually write A << {x} as A << x.
This will give us an awful lot of infinitesimals, because it means we can pick any decreasing sequence of positive numbers and there will be a smaller one. e.g. there is some x_0 such that 0 < x < 1/n for every n. Then there is some x_1 such that 0 < x_1 < x_0^n for every n, x_2, etc. then there is y with 0 < y < x_n for every n, etc. You get the idea. There are a lot of infinitesimals.
Given a bit of faffing we can turn the above argument into the following:
Proposition 2:
There is a strictly decreasing sequence of positive infinitesimals of length aleph_1.
I think this will be important later, but I'm not sure.
Also, note that proposition 1 means that there are no convergent sequences in the order topology of K except those which are eventually constant. Proof:
Suppose we have a sequence x_n -> x such that for all n, x_n != x. Let A = { x_n : x_n < x} and B = { x_n : x_n > x}. Then we can find s with A << s < x and t with x < t << B. Then the sequence x_n never enters the neighbourhood (s, t) of x, so does not converge to x. Contradiction.

We can restate proposition 1 as:

Proposition 1′:

Let K be an ultrapower and A, B subseteq K be countable with A << B. Then there exists x with A << x << B.
This motivates the following definitions:
Let X be an ordered set and A, B be subsets which are not both empty. Then (A, B) forms a pregap if A << B.
If A << x << B then we say x interpolates the pregap (A, B).
A pregap is a gap if it has no interpolating element. It is countable if both A and B are countable.

So, one more restatement of the proposition:

Proposition 1”:

Let K be an ultrapower. K has no countable gaps.

We’ll turn this into a definition. An ordered set is an eta_1 set if it has no countable gaps.

Also, an ordered set is an alpha_1 set if every subset of it has a countable order dense subset. (A <= X is order dense if for every x, y in X there is an a in A with x < a < y). e.g. R is an alpha_1 set. The intuition here being that alpha_1 sets are small, eta_1 sets are large. We will mostly be considered with ordered fields which are eta_1 as ordered sets. Ultrapowers were our first example of such, but we'll see a lot of other examples later. It will apparently turn out to be very important to study what kinds of gaps our ordered field extensions have. I'll include more on that next time.