Processes as numbers: part 2

Recap: it would be nice to add a number such as $\infty$ as an "infinite number", and then to determine the behaviour of a sequence $(a_{n})$ as $n$ "goes to infinity" just look at $a_{\infty}$ .

Adding an "infinite number" to a system in this way is similar to the process when a system is "completed" by adding all values of limits (such as how each irrational real number is a limit of rationals and added to $ℚ$ to form $ℝ$ ). The problems with the example of $a_{\infty}$ are that (1) we need to define the sequence's value at $\infty$ and (2) there are, in fact, many ways of "going to infinity" and each should get a different infinite number like $\infty$ .

With that said, Robinson's method for constructing nonstandard numbers does have a lot of similarities with "completions" in mainstream mathematics.

1. The ultrafilter construction

This section covers the ultrafilter construction: it is rather concrete and straightforward. We start with a sequence $(a_{n})$ and want to investigate its behaviour "at infinity".

A real-valued process is a sequence of real numbers $b_{n}$ We will make all real valued processes into numbers with the help of an ultrafilter. An ultrafilter $D$ is a subset of $P (X)$ where $X$ is an infinite indexing set (here we take $X = ℕ$ ). The ultrafilter $D$ must satisfy certain axioms.

$\emptyset \notin D$
$A \in D$ and $B \in D$ implies $A \cap B \in D$
$A \in D$ and $B \supseteq A$ implies $B \in D$
for all $A \subseteq ℕ$ , $A \in D$ or $X ∖ A \in D$

Think of $D$ as specifying the large subsets of $X$ . (They are so large that the intersection of any two large sets is large.)

The easiest example of an ultrafilter is the principal ultrafilter for some $a \in X$ , defined as $D_{a} = \{A \subseteq X: a \in A\}$ . An ultrafilter $D$ is nonprincipal if it is not of this type.

For an example of a nonprincipal ultrafilter one must work a bit harder. Start with $Cofin = \{A \subseteq X: X ∖ A is finite\}$ . This is a filter, i.e. satisfies the first three axioms. Now using Zorn's lemma extend $Cofin$ to a maximal filter in $P (X)$ . This will be an ultrafilter: the mathematical work is done when given a filter $D$ not containing $A$ nor $X ∖ A$ one observes that

D' = \{B: \exists C \in D B \supseteq A \cap C\}

is a filter properly extending $D$ . To see that an ultrafilter extending $Cofin$ is not principal just note that it contains $X ∖ \{a\}$ for all $a \in X$ .

We now fix an ultrafilter $D$ for $X = ℕ$ extending $Cofin$ and look at real-valued processes $(b_{n})$ . The trick is to say two such sequences $(b_{n})$ and $(c_{n})$ are equivalent (essentially equal) if the set of indices where they are equal is "large" i.e. in $D$ .

(b_{n}) \sim (c_{n}) \Leftrightarrow \{n: b_{n} = c_{n}\} \in D

Exercise: show this is an equivalence relation.

I will notate the equivalence class of the sequence $(b_{n})$ by $b_{D}$ .

The new extended set of numbers, denoted $Π_{D} ℝ$ , or $* ℝ$ , is the set of real-valued processes $(b_{n})$ modulo this equivalence. It includes all normal real numbers because each $r \in ℝ$ gives rise to a constant process $b_{n} = r$ , all $n$ , whose equivalence class we identify with $r$ .

Exercise: show that all distinct constant real-valued processes are inequivalent under the equivalence relation.

For an example of an infinite number, consider the process given by $b_{n} = n$ . This is not equal to any $r$ since $\{n: n \neq r\}$ is in $Cofin$ . The number corresponding to the original $(a_{n})$ is typically a new number too. To picture where this arises we need to put some new structure on $* ℝ$ .

b_{D} < c_{D} \Leftrightarrow \{n: b_{n} < c_{n}\} \in D

b_{D} + c_{D} = d_{D} where d_{n} = b_{n} + c_{n}

b_{D} \times c_{D} = d_{D} where d_{n} = b_{n} \times c_{n}

Other operations such as $-$ and $||$ can be defined in an analogous way. In all cases, these definitions agree with the usual one on $ℝ$ because of the identification of $r \in ℝ$ with a constant sequence and $ℕ \in D$ .

Exercise. For $b_{n} = n$ show that $b_{D} > r$ for all $r \in ℝ$ . (We write this as $b_{D} > ℝ$ .)

The system $* ℝ$ contains infinitesimal numbers too, such as $b_{D}$ where $b_{n} = 1 / n$ . It is easily seen that $0 < b_{D} < r$ for all positive $r \in ℝ$ . (Exercise.)

Here is an important definition: write $a \approx b$ when $|a - b|$ is zero or infinitesimal.

Proposition. For $a \in * ℝ$ , $a \approx r$ for at most one $r \in ℝ$ .

When $a \approx r$ for some $r \in ℝ$ we say $a$ is finite and write $r = st a$ .

Theorem. For the sequence $(a_{n})$ and $l \in ℝ$ we have, $a_{n} \to l$ iff $a_{D} \approx l$ for all ultrafilters $D \supseteq Cofin$ .

One direction goes by assuming $a_{n} \to l$ and showing $|a_{D} - l| < ε$ for each positive $ε \in ℝ$ . This is rather standard in the literature. The other direction (which is less often presented) goes by assuming $l$ is not a limit and so for some $ε > 0$ in $ℝ$ there are $n_{i} \in ℕ$ with $n_{0} < n_{1} < \dots$ and $|a_{n_{i}} - l| > ε$ for $i \in ℕ$ . Then there is an ultrafilter $D \supseteq Cofin$ containing $\{n_{0}, n_{1}, \dots\}$ and

\{n \in ℕ: |a_{n} - l| > ε\} \supseteq \{n_{0}, n_{1}, \dots\} \in D

showing $|a_{D} - l| > ε$ for this $D$ .

2. A few more definitions, and tidying up

We haven't quite got to where we want to be: we have infinite numbers such as the number $b_{D}$ for the process $b_{n} = n$ but we haven't made sense of $a_{b_{D}}$ yet, and the characterisation of convergence involves a quantifier over ultrafilters $D$ .

The first is easily fixed. Since $a_{n}$ is a function it extends naturally to the nonstandard universe. First

The set $* ℕ \subseteq * ℝ$ is the set of $b_{D}$ where $\{n: b_{n} \in ℕ\} \in D$ . For any such $b_{D}$ we set

a_{(b_{D})} = c_{D} where c_{n} = a_{(b_{n})}

Note that only the $b_{n} \in ℕ$ matter because there is a $D$ -large set of such indices $n$ . We could set $a_{(b_{n})} = 0$ for all other $n$ , or indeed anything else.

Now we have $a_{x}$ for any $x \in * ℕ$ , including infinite such $x$ .

Theorem. $a_{n} \to l$ iff $a_{x} \approx l$ for all infinite $x \in * ℕ$ .

For one direction, suppose $a_{n} \to l$ and $x = x_{D} \in * ℕ$ . then for each $ε > 0$ there is $N \in ℕ$ so that $\{n: |a_{n} - l| < ε\} \supseteq \{n: n > N\}$ , and as $x_{D}$ is infinite, $\{n: x_{n} > N\} \in D$ . Therefore $\{n: |a_{n} - l| < ε\} \in D$ .

For the other, suppose $a_{n} \to l$ is false and take $ε > 0$ and natural numbers $x_{0} < x_{1} < x_{2} < \dots$ with $|a_{(x_{i})} - l| > ε$ . So $x_{D}$ is infinite (because the sequence $(x_{n})$ is increasing) and $|a_{(x_{D})} - l| > ε$ .

Remark. It is certainly remarkable that the notion $a_{n} \to l$ (which involves three quantifiers, $\forall ε > 0 \exists N \in ℕ \forall n > N \dots$ ) is equivalent to " $a_{x} \approx l$ for all infinite $x \in * ℕ$ " which involves only one quantifier, quantifying over infinite $x \in * ℕ$ . Of course the "infinite" $x$ is doing some work as is $\approx$ . Note that neither of these two key notions are defined in the "usual" way by saying some set of indices is in $D$ . You could ask, given that $\forall ε > 0 \exists N \in ℕ \forall n > N (|a_{n} - l| < ε)$ gives a function $f$ with $\forall ε > 0 \forall n > f (ε) |a_{n} - l| < ε$ we should be able to obtain a similar function from the nonstandard characterisation. Indeed we can: take

f (ε) = least N \in * ℕ such that \forall n \in * ℕ (n > N \to |a_{n} - l| < ε)

It is easy to check that this definition gives finite $f (ε)$ for positive standard real $ε$ .