# Processes as numbers: part 2

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

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 $a ∞$ are that (1) we need to define the sequence's value at $∞$ and (2) there are, in fact, many ways of "going to infinity" and each should get a different infinite number like $∞$.

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.

• $∅ ∉ D$
• $A ∈ D$ and $B ∈ D$ implies $A ∩ B ∈ D$
• $A ∈ D$ and $B ⊇ A$ implies $B ∈ D$
• for all $A ⊆ ℕ$, $A ∈ D$ or $X ∖ A ∈ D$

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 $a ∈ X$, defined as $D a = A ⊆ X a ∈ A$. 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 . 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

$D ′ = B ∃ C ∈ D B ⊇ A ∩ C$

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 $a ∈ X$.

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 $D$.

$( b n ) ∼ ( c n ) ⇔ n b n = c n ∈ D$

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 $r ∈ ℝ$ 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 $n n ≠ r$ 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 $* ⁢ ℝ$.

$b D < c D ⇔ n b n < c n ∈ D$

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 $r ∈ ℝ$ with a constant sequence and $ℕ ∈ D$.

Exercise. For $b n = n$ show that $b D > r$ for all $r ∈ ℝ$. (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 $r ∈ ℝ$. (Exercise.)

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

Proposition. For $a ∈ * ⁢ ℝ$, $a ≈ r$ for at most one $r ∈ ℝ$.

When $a ≈ r$ for some $r ∈ ℝ$ we say $a$ is finite and write $r = st ⁡ a$.

Theorem. For the sequence $( a n )$ and $l ∈ ℝ$ we have, $a n → l$ iff $a D ≈ l$ for all ultrafilters $D ⊇ Cofin$.

One direction goes by assuming $a n → l$ and showing $a D - l < ε$ for each positive $ε ∈ ℝ$. 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 $n i ∈ ℕ$ with $n 0 < n 1 < …$ and $a n i - l > ε$ for $i ∈ ℕ$. Then there is an ultrafilter $D ⊇ Cofin$ containing $n 0 , n 1 , …$ and

$n ∈ ℕ a n - l > ε ⊇ n 0 , n 1 , … ∈ D$

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 $* ⁢ ℕ ⊆ * ⁢ ℝ$ is the set of $b D$ where $n b n ∈ ℕ ∈ D$. For any such $b D$ we set

Note that only the $b n ∈ ℕ$ 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 $x ∈ * ⁢ ℕ$, including infinite such $x$.

Theorem. $a n → l$ iff $a x ≈ l$ for all infinite $x ∈ * ⁢ ℕ$.

For one direction, suppose $a n → l$ and $x = x D ∈ * ⁢ ℕ$. then for each $ε > 0$ there is $N ∈ ℕ$ so that $n a n - l < ε ⊇ n n > N$, and as $x D$ is infinite, $n x n > N ∈ D$. Therefore $n a n - l < ε ∈ D$.

For the other, suppose $a n → l$ is false and take $ε > 0$ and natural numbers $x 0 < x 1 < x 2 < …$ 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 $a n → l$ (which involves three quantifiers, $∀ ε > 0 ∃ N ∈ ℕ ∀ n > N …$) is equivalent to "$a x ≈ l$ for all infinite $x ∈ * ⁢ ℕ$" which involves only one quantifier, quantifying over infinite $x ∈ * ⁢ ℕ$. Of course the "infinite" $x$ is doing some work as is $≈$. 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 $∀ ε > 0 ∃ N ∈ ℕ ∀ n > N ( a n - l < ε )$ gives a function $f$ with $∀ ε > 0 ∀ n > f ⁡ ( ε ) a n - l < ε$ we should be able to obtain a similar function from the nonstandard characterisation. Indeed we can: take

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