Hyperreals

Between Perl and Ruby I try to find some time for math. As time passes I am more and more slow at digesting a page of stuff, but math is so beautiful! Reading a math book needs concentration and time to do exercises and think about what you learn to absorb it, which often gets interrupted by Real Life™ unfortunately.

I have founded an excellent book about a topic I wanted to read about sometime: Hyperreals. It is less than 300 pages long so I'll try to finish this one. Moreover the topic touches two areas I really enjoyed in the faculty: Mathematical Analysis and Mathematical Logic. (Well, to be honest I think I enjoyed everything except Statistics and Probability.)

A bit of History

Leibniz, Newton, and others in the 17th and 18th centuries use the ill-defined concept of infinitesimals. They base their math on these mysterious quantities, and do unknown things with them such as arithmetic, discarding terms when they are too small compared to other numbers, etc. Not very sound, but you know, the standard of rigour in math is a function of time :-).

So, epsilons and deltas weren't used originally. That's a formalism introduced in the 19th century by German mathematicians because at that time there were ambiguous definitions in the fundamentals of calculus and it was hard to move on in a confident way. Karl Weierstrass introduced the modern notion of rigour into Mathematical Analysis.

After the "refactoring" of the 19th infinitesimals were no longer used, everything was rigorously defined and proved in terms of epsilons and deltas, limit, continuity, differentiation, etc.

The Compactness Theorem

We jump now to the 20th century and the modern development of Mathematical Logic.

Without getting too specific, given a set of axioms we say that such mathematical entity is a model of them if the entity satisfies them. For example we say that the integers are a model of the axioms of a commutative ring. They are not a model of the axioms of a field, because there are non-null elements in Z that have no multiplicative inverse in Z.

Alright, there's a fundamental result in Mathematical Logic called Compactness Theorem that more or less says that if any finite subset of a (possibily infinite) set of axioms has a model, then the entire set of axioms has a model. We'd need to know Mathematical Logic to give a precise theorem but that idea is enough.

Take the axioms of an ordered field and add to them these:

Axiomn = there exists ε such that 0 < ε < 1/n.

Now, R and Q are models of any finite subset of these axioms, and it follows from the compactness theorem that there's a model of the entire set of axioms. That's an ordered field that has numbers strictly greater than zero and strictly less than 1/n for all n in N. There you have it, the infinitesimals back on scene.

A Construction Of The Hyperreals

Besides, that is not waving hands, the compactness theorem has a proof that actually constructs the model using a structure which is very used in logic called an ultraproduct . Reals can be constructed from the rationals using Dedekind cuts or equivalence classes of Cauchy sequences. I was lucky to see that construction in detail in my first year in the faculty. The Hyperreals are constructed as equivalence classes of real sequences modulus a nonprincipal ultrafilter on N. Two sequences are equivalent iff the set of indexes where their elements are equal belongs to the ultrafilter. The book motivates and goes through this construction. The existence of such a nonprincipal ultrafilter is not constructive though, it is proven using Zorn's Lemma.

The result is an ordered field, *R, that embeds the reals, has infinitesimals, and has as a consequence infinite numbers, greater than any real number: they are the multiplicative inverses of the infinitesimals the axioms of field guarantee. Properties of hyperreals are easily proven to hold in R thanks to the transfer principle, which informally explained says that any appropriately formulated statement is true of *R if and only if it is true of R. So you can work in *R and transfer the result to R by magic of the transfer principle. Of course the transfer principle has a precise meaning explained on Chapter 4.

A very important property of R for which the transfer principle cannot be applied is that R is complete, that is, every Cauchy sequence of real numbers converges in R. This had to happen because it is known that all complete ordered fields are isomorphic to R. So completeness is something we do not have in *R.

Non-Standard Analysis

This gave the chance to rewrite analysis using these contemporary infinitesimals with rigour. And this is what it is known as Non-Standard Analysis, which was introduced by the father of these ideas, Abraham Robinson.

The result is a beautiful and intuitive theory, where you can prove things talking about being infinitely close.