Comments on Edward Nelson’s “internal Set Theory: a New Approach to Nonstandard Analysis”

Do infinitesimals exist? This question arises naturally in calculus when one wants to write f(x+Δx) f(x) + f ′(x)Δx. This equation should hold for infinitesimal Δx where the notation a b means that a− b is infinitesimal. Much calculus can be done without bothering to define infinitesimal and using the equation above as a working definition of f ′(x). For mathematicians, of course, this is not a satisfactory starting point for a theory. We are trained that we cannot answer a question such as, “Do infinitesimals exist?” without first defining the term infinitesimal. One of the achievements of nineteenth century mathematics was to give a rigorous foundation for the differential calculus using the idea of the limit. This is what we teach in undergraduate real analysis courses usually after the students have some experience and intuition in calculus. A necessary step in this program is to describe precisely what a “real” number is. Typically, one does this by giving the properties that we believe the real numbers have, e.g., a complete ordered field, and then constructing a set of numbers with such properties using Cauchy sequences or Dedekind cuts. The vast majority of students (and probably of mathematicians) consider the construction just a formality. Clearly, the real numbers exist and have these properties. Indeed, many courses in elementary analysis choose not to construct the reals but rather to take the existence of an ordered field as given. This is reasonable: we are implicitly assuming such an object exists, otherwise, why are we studying it? Unfortunately, the nineteenth century rigorization of calculus was not completely rigorous because it did not answer fundamental questions about set theory and logical structures. Today the most common set theory is Zermelo–Fraenkl (ZF) usually with the axiom of choice added (ZFC). While techincally this is the starting point for proofs of almost all of mathematics, few could actually give the axioms of ZF. It is also not known whether or not the axiom system ZFC is consistent! Of course, this does not really disturb most mathematicians since they “know” that