All things ultra

I’ve personally always been fascinated by the notion of an ultraproduct. Before I came to know of the construction, I guess it could’ve been because it kept popping up and was “unknown territory” – and the cool sounding name, I must admit. However, after actually knowing what the construction is all about, it doesn’t fascinate me any less, which is largely due to its powerful applications. I’ll here try to define the notion of ultrafilters, ultraproducts and ultrapowers as well as showing a particular application, namely a rigorous account of infinitesimals in a branch of mathematics called non-standard analysis. But before getting to that, we need to define a filter and an ultrafilter, which will be key concepts in the ultraproduct construction.

Definition 1. Let x be a nonempty set. Then \mathcal{F}\subseteq\mathcal{P}(x) is called a filter over x if it satisfies that

  • \emptyset\notin\mathcal{F}
  • x\in\mathcal{F}
  • \forall a,b\subseteq x: a\in\mathcal{F}\land a\subseteq B\Rightarrow b\in\mathcal{F}
  • \forall a,b\subseteq x: a,b\in\mathcal{F}\Rightarrow a\cap b\in\mathcal{F}

This somehow cryptic definition can make intuitive sense if one replaces the notion of being an element of \mathcal{F} as being “nearly everything”. The first three conditions thus say that \emptyset is not nearly everything, x is nearly everything and if a is nearly everything, and b is bigger than a, then b is also nearly everything. The last condition states that if a,b are both nearly everything, then what they have in common is also nearly everything, which is the only semi-sketchy condition about filters, but nevertheless important.

We now come to the notion of an ultrafilter:

Definition 2. A filter \mathcal{F} over some set x is called an ultrafilter if for every subset a\subseteq x, either a\in\mathcal{F} or x\backslash a\in\mathcal{F}.

An example of an ultrafilter is if p\in x, then the principal ultrafilter over x is defined as \{a\subseteq x\mid p\in a\}. Another example of a filter is the Fréchet filter (also called the cofinite filter) over \omega, defined as \{a\subseteq\omega\mid|\omega\backslash a|<\omega\}. It is noted that the Fréchet filter is not in fact an ultrafilter, since we can find a subset of omega satisfying that both it and its complement isn’t in the filter; namely the even and odd numbers (check!).

Now, as a last definition before getting started on the ultraproducts, is how a product of sets is even defined:

Definition 3. Let \langle a_n\mid n<\omega\rangle be a sequence of sets. We then define the product of the a_n‘s to be the set of functions f with dom(f)=\omega and f(n)\in a_n for every n<\omega. Symbolically:

\prod_{n<\omega}a_n:=\{f\mid f\text{ function }\land dom(f)=\omega\land\forall n<\omega: f(n)\in a_n\}

Now, with all those definitions down, let’s start constructing an ultraproduct. First of all, we take some sequence \langle a_n\mid n<\omega\rangle and define P:=\prod_{n<\omega}a_n as their product. Now we let \mathcal{F} be an ultrafilter over \omega. So now we got the product P and the ultrafilter \mathcal{F}, which we want to link together. This is done through a relation \sim on P, defined by

f\sim g\Leftrightarrow\{n<\omega\mid f(n)=g(n)\}\in\mathcal{F}

It can be showed that this relation is in fact an equivalence relation (check!). Since this is the case, we can form the quotient space

a:=P/\sim=\{[f]_\sim\mid f\in P\}.

Now assume that to each a_n there is some relation <_n\subseteq a_n\times a_n. Then we can define a relation <\subseteq a\times a by

[f] < [g]\Leftrightarrow \{n<\omega\mid f(n) <_n g(n)\}\in\mathcal{F},

which can be shown to be well-defined (what needs to be checked?). Finally, \langle a,<\rangle is then called the ultraproduct of the sequence \langle\langle a_n,<_n\rangle\mid n<\omega\rangle by \mathcal{F} and usually denoted by

\prod_{n<\omega} \langle a_n,<_n\rangle/\mathcal{F}.

If all the a_n‘s and <_n‘s happen to be the same, the construction is called an ultrapower instead.

An example: hyperreal construction

To give an example of all this, as well as to demonstrate that this seemingly odd construction has powerful implications, we will here take the ultrapower of \langle\mathbb{R},<\rangle, where < is the usual ordering on the real numbers. This ultrapower will then give us what is called the hyperreals \langle\Bbb{R}^*,<^*\rangle. So let’s construct them!

First of all, since all the a_n‘s are the same, the product is merely the set of all functions from \omega to \mathbb{R}:

\prod_{n<\omega}\langle\mathbb{R},<\rangle = ^\omega\mathbb{R}.

Now let \mathcal{F} be a nonprincipal ultrafilter over \omega. We now define the equivalence relation \sim\subseteq{^\omega\mathbb{R}}\times{^\omega\mathbb{R}} as before, by for all f,g\in{^\omega\mathbb{R}}:

f\sim g\Leftrightarrow \{n<\omega\mid f(n)=g(n)\}\in\mathcal{F}.

We thus form the quotient space \mathbb{R}^*:=^\omega\mathbb{R}/\sim and furthermore define the relation <^*\subseteq\mathbb{R}^*\times\mathbb{R}^* by setting

[f]_\sim<^*[g]_\sim\Leftrightarrow \{n<\omega\mid f(n)<g(n)\}\in\mathcal{F}.

Now \langle\mathbb{R}^*,<^*\rangle is the hyperreals. The special thing about this structure is, that it contains the (positive/negative) infinite as a proper number, as well as having infinitesimals as members. This can be shown by firstly noting that the reals can be embedded into this structure, by noting that the function (written in lambda notation):

\lambda a. [n\mapsto a]_\sim:\langle\mathbb{R},<\rangle\to\langle\mathbb{R}^*,<^*\rangle (in regular notation f(a)=[n\mapsto a]_\sim)

is a monomorphism since a<b\Rightarrow [n\mapsto a]_\sim<^*[n\mapsto b]_\sim (check!). So there’s a copy of the reals in the hyperreals. But there’s other elements in the hyperreals as well, for instance [n\mapsto n]_\sim and [n\mapsto -n]_\sim, which satisfies that for every real number a it holds that

[n\mapsto -n]_\sim<^*[n\mapsto a]_\sim<^*[n\mapsto n]_\sim,

meaning that these two elements correspond to the notion of (positive/negative) infinity. If we furthermore let a,b\in\mathbb{R} with a<b, then we have

[n\mapsto a]_\sim<^*[n\mapsto a+1/n]_\sim<^*[n\mapsto b]_\sim,

meaning that there are numbers infinitely close to a, i.e. that it is strictly greater than a and strictly smaller than every real number strictly greater than a (if a=0, then we let the function n\mapsto a+1/n be 0 when n=0). This whole shibang leads up to a whole new treatment of real analysis, without the use of any epsilons and deltas! Indeed, one can formulate equivalent notions of things like convergence and continuity with infinitesimals instead of working with limits – this interesting area of mathematics is called non-standard analysis.

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