Doing Calculus on the Rationals (with the help of Nonstandard Analysis)

Nonstandard Analysis is usually used to introduce infinitesimals into the real numbers in an attempt to make arguments in analysis more intuitive.

The idea is that you construct a superset which contains the reals and also some infinitesimals, prove that some statement holds of , and then use a general “transfer principle” to conclude that the same statement holds of .

Implicit in this procedure is the idea that is the real world, and therefore the goal is to prove things about it. We construct a field with infinitesimals, but only as a method for eventually proving something about .

We can do precisely the same thing with instead of with . But, in Weak Theories of Nonstandard Arithmetic and Analysis, Jeremy Avigad observed that if we don’t care about transferring the results back down to , then we can get all the basic results of calculus and elementary real analysis just by working with , and without ever having to construct the reals.

Let me first differentiate two approaches to nonstandard analysis. The first is the one I mentioned above, where you actually construct a field (although you need the axiom of choice to do it). This is done entirely within ordinary mathematics. Call this the semantic approach.

Another approach is the axiomatic approach. A good example of this is Edward Nelson‘s internal set theory. In this approach, you take an ordinary axiomatization of some part of mathematics (for example, ZFC), introduce a new predicate for being “standard” or “normal-sized”, and some axioms saying that there exist things which are not standard and how these things relate to everything else. In the usual situation, a sentence which does not contain the predicate “standard” is provable in the new theory iff it’s provable in the old theory. (This is the case with IST and ZFC.)

The axiomatic approach is the approach we’ll take here. We’ll let our language consist of a function symbol for each primitive recursive function and relation, together with a predicate and a constant . Our axioms will be the following:

1. If is a true (in the natural numbers) first-order -sentence that does not include the new predicate , then we take as an axiom.
2. We take as an axiom.
3. We take as an axiom.
4. We take to be an axiom for each -ary primitive recursive function .

The interpretation of our sentences is that we are now quantifying over a domain which includes infinitely large natural numbers (of which is an example) and that the predicate picks out those which are normal-sized. However, since we are working within the axiomatic system, I will still refer to the domain we are quantifying over as .

Within the system, construct and from as usual. We make the following definitions:

We say that an natural number is unbounded if it is not standard (i.e., if ). We say that an integer is unbounded if is unbounded. We say that a rational is unbounded if the closest integer to it is unbounded.

Furthermore, we say that a rational number is infinitesimal if it equals 0 or if is unbounded. We say that and are infinitely close, written , if is infinitesimal.

Let be the set of rationals which are not infinite. We can now do analysis on . First of all, we can define continuity in a natural way: We say that is continuous if whenever , .

We have the intermediate value theorem for : If and and is continuous, then there is a such that . Proof: Recall that is a natural number. Let be the maximum natural number less than such that . (This is possible because there are only finitely many natural numbers less than any natural number, including !) But then must be infinitely close to , since by continuity .

We can also prove that any continuous function on attains a maximum (up to ) by essentially the same means: just consider the for which is a maximum, which is again possible considering that there are only finitely many .

Turning to differentiation, we may define if for all non-zero infinitesimals ,

(Note that the derivative is actually defined only up to .) We can then prove that the derivative of is by letting be an arbitrary infinitesimal, expanding , dividing by , and noting that what results is plus an infinitesimal.

Avigad notes that we may continue by defining , , and by taking an unbounded partial sum of the Taylor expansions, and that this is sufficient to prove all the basic properties. He also cites an easy proof in this setting of the Cauchy-Peano theorem on the existence of solutions to differential equations.