When I was an undergraduate, I remember being very struck by some of the early results in the class I was taking on abstract algebra. Of course, I was eventually very struck by the results from Galois theory when we got there, but in the early parts of the class I was struck by the results proving the existence of the algebraic closure of a field, and proving the existence of a field of fractions for every integral domain. In particular, these seemed to me to validate the use of the complex numbers (once the reals were given) and the rational numbers (once the integers were given). I was still vaguely dissatisfied that we hadn’t yet had a proof of the existence of the integers, but I became happier when I saw the definition of the natural numbers as the smallest set containing 0 and closed under the successor operation, especially because this let proof by induction be a theorem rather than an axiom.
However, earlier this week (in conversation with Zach Weber, while visiting Sydney), I started realizing what I should have realized long ago, which is that these theorems really can’t be doing as much work in justifying our use of the various number concepts as I had thought when I was younger. Of course, these theorems are quite useful when talking about abstract fields or rings, but when we’re talking about the familiar complex, real, rational, and integer numbers, it’s no longer clear to me that these theorems add anything whatsoever. After all, what these theorems show is just that, by using some fancy set-theoretic machinery of ordered pairs and equivalence classes, we can create a structure that has all the properties of a structure that we already basically understood. Perhaps in the case of the complex numbers this mathematical assurance is useful (though even there we already had the simple assurance in the form of thinking of complex numbers as ordered pairs of reals, rather than as polynomials over R modulo the ideal [x2+1]), but for the rationals and negative numbers, our understanding of them as integers with fractional remainder, or as formal inverses of positive numbers, is already sophisticated enough to see that they’re perfectly well-defined structures, even before we get the construction as equivalence classes of ordered pairs of integers or naturals.
But this is all a sort of prelude to thinking about the two more famous set-theoretic reductions, that of the reals to Dedekind cuts (or Cauchy sequences) of rationals, and that of the naturals to the finite von Neumann ordinals. Unlike the others, I think the Cauchy and Dedekind constructions of the reals are quite useful – before their work, I think the notion of real number was quite vague. We knew that every continuous function that achieves positive and negative values should have a zero, but it wasn’t quite clear why this should be so. Also, I think intuitively there remained worries about whether there could be a distinct real number named by “.99999…” as opposed to the one named by “1”, not to mention the worries about whether certain non-convergent series could be summed, like 1-1+1-1+1….
But for the reduction of the naturals to the von Neumann ordinals, I think it’s clear that this should do no work in explicating the notion at hand. To prove that enough von Neumann ordinals exist to do this work, you already need a decent amount of set theory. (John Burgess’ excellent book Fixing Frege does a good job investigating just how much set theory is needed for this and various other reductions.) And while some of the notions involved are basic, like membership and union, I think the concept of sets of mixed rank (for instance, sets that have as members both sets, and sets of sets) already strains our concept of set much more than any of this can help clarify basic notions like successor, addition, and multiplication. One might even be able to make a case that to understand the relevant formal set theory one must already have the concept of an ordered string of symbols, which requires the concept of finite ordering, which is basically already the concept of natural numbers!
In some sense, this was one project that Frege was engaged in, and his greatest failure (the fact that his set theory was inconsistent) shows in a sense just how unnecessary this project was. At least to some extent, Frege’s set theory was motivated by an extent to show the consistency of Peano arithmetic, and clarify the concept of natural number. However, when his explanation failed, this didn’t undermine our confidence in the correctness of Peano arithmetic. The same thing would be the case if someone today were to discover that ZFC was inconsistent – most of the mathematics that we today justify by appeal to ZFC would still stand. We wouldn’t abandon Peano arithmetic, and I think we wouldn’t even abandon most abstract algebra, geometry, analysis, and the like, except perhaps in some cases where we make strong appeals to the Axiom of Choice and strange set-theoretic constructions.
Of course, Frege’s actual attempted reduction of the number concepts to set theory would have been a very nice one, and could help explain what we mean by number, because he reduced each number to the collection of all sets with that many elements. However, modern set theory suggests that no such collections exist (except in the case of the number 0), and the proposed reductions are much less illuminating.
So I wonder, what role do these proofs play, that demonstrate the existence of structures that behave like the familiar natural numbers, integers, rationals, reals and complex numbers? I’ve suggested that in the case of the reals it may actually do important work, but I’m starting to be skeptical of most of the other cases.