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 to form ). 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 = ). The ultrafilter D must satisfy certain axioms.

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 Cofin = A 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 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 = 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 , or * , is the set of real-valued processes ( b n ) modulo this equivalence. It includes all normal real numbers because each 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 * .

b D < c D n b n < c n D b D + c D = d D  where  d n = b n + c n b D × c D = d D  where  d n = b n × 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 because of the identification of r with a constant sequence and D .

Exercise. For b n = n show that b D > r for all r . (We write this as b D > .)

The system * 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 . (Exercise.)

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

Proposition. For a * , a r for at most one r .

When a r for some r we say a is finite and write r = st a .

Theorem. For the sequence ( a n ) and l 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 . 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 there are n i with n 0 < n 1 < and a n i - l > ε for i . Then there is an ultrafilter D Cofin containing n 0 , n 1 , and

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 * * is the set of b D where n b n 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 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 * , including infinite such x .

Theorem. a n l iff a x l for all infinite x * .

For one direction, suppose a n l and x = x D * . then for each ε > 0 there is 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 ) is equivalent to " a x l for all infinite x * " which involves only one quantifier, quantifying over infinite x * . 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 ( 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

f ( ε ) = least  N *  such that  n * ( n > N a n - l < ε )

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