A REVOLUTION IN THE FOUNDATIONS OF MATHEMATICS?

The standard contemporary view of the foundations of mathematics and of the role of logic in it is well known and firmly entrenched. The ground floor of the edifice of mathematics is on this view our basic logic, that is to say first-order logic, a.k.a. quantification theory or lower predicate calculus. It has various desirable features, such as completeness, compactness, Löwenheim–Skolem property and the validity of the separation theorem. Indeed, it is the strongest possible logic in the sense of abstract (model-theoretical) logic that has the pleasant properties of compactness and Löwenheim–Skolem property, assuming only the usual behavior of propositional connectives plus a few plausible structural properties. This is what the famous theorem of Lindström’s shows.1 This theorem seems to assign a special position to ordinary first-order logic. At the same time, it is a kind of impossibility theorem, showing that we cannot hope to strengthen first-order logic without losing some of its desirable properties. First-order logic was first formulated explicitly by Frege as a part of his more comprehensive Begriffsschrift. Frege had to go further, however. His logic is not first-order, but higher-order.2 In other words, Frege’s logic allows quantification not only over individuals (particulars), but also over higher-order entities, such as functions and other concepts applying to individuals. The problems caused by this transgression beyond first-order concepts will be discussed later in this paper. It is sometimes said that the idea of a freestanding first-order logic was not known to Frege and that it crystallized only later, for the first time apparently in Hilbert’s and Ackermann’s 1928 textbook (Moore 1988). Maybe so. But even if Frege should have entertained the idea of pure first-order logic, he would have had plenty of good prima facie reasons to go beyond what is in our days known as first-order logic. The most basic reason is that ordinary first-order logic does not suffice for mathematics. Another reason is that it is not even self-sufficient. The first of these two failures is illustrated by the fact that several of the most basic concepts of