What's Wrong with ZFC?

A discussion of reasons for considering alternatives to ZFC as a foundation for mathematics.

During the first half of this century classical mathematics, as constructed within the foundational framework provided by ZFC was challenged primarily, but not effectively, by intuitionist doctrines. (We take ZFC here to be representative of those first order theories based on the iterative conception of set which could equally well have served as a foundation for the mathematics of the period.)

In the second half a wider range of influences from philosophy, mathematics and computer science have increased pressure on the iterative conception of sets as the foundation for modern mathematics. The most signficant of these pressures comes from Computer Science. Many academic computer scientists with theoretical inclinations are drawn to the power of abstraction provided by category theoretic methods, while category theoreticians themselves strain against the limitations on abstraction imposed by classical set theory. At the same time arguments for constructive mathematics have been plausibly aimed at computer scientists. Arguments that these are more relevant and practical in computing than classical methods are accepted enthusiastically and without scrutiny.

Academics will persue research provided only that they find the material interesting and that some else finds it fundable. Funding agencies in turn will often operate by peer review aiming primarily to fund the most able researchers whose opinions about the merits of various avenues of research are definitive.

In this context a philosophical interest in the reasons why particular lines are followed may be difficult to satisfy.

We will consider criticisms of classical foundations arising from the following general sources:

Constructive Influences
There are many approaches to mathematics which are broadly constructive. Many of these systems are described in Foundations of Constructive Mathematics, but the years since the publication of this book have seen significant further advances. The challenge posed by these systems is both to the foundations and often also to mathematics itself
Category Theory
Category Theory suffers from some of the limitations inherent in the iterative conception of set, and at the same time because of its degree of abstraction appears to provide a new way of approaching all other branches of mathematics. For this reason some category theorists have suggested that category theory itself should in some way supplant ZFC as a foundation for mathematics. In their attempts to develop categorical approaches to set theory category theorist have developed ideas which have subsequently been shown to connect closely with intuitionism. The convergence of these two sources of influence shows signs of creating a new conventional wisdom within theoretical computer science about the foundations appropriate to computer science and mathematics.
Other Factors
Even when constructive tendencies are discounted problems arise from the mechanisation of mathematics and other factors. We will separately address some motives for changes to foundational systems which arise independently of constructive or category theoretic motivations.

R.© Roger Bishop Jones; Created: 1995/3/5; Modified: 1995/3/5