Set theory and the continuum

pp. 112-144

in: Janet Folina, Poincaré and the philosophy of mathematics, Basingstoke, Palgrave Macmillan, 1992

Abstract

In the same way that Poincaré's theory of arithmetic intuition is exhibited in his negative arguments against the logicist, his theory of geometric intuition (of continuity) is exhibited in his negative arguments against the set theorist. Thus we must examine his attack on set theory in order to uncover the details of his theory of geometric intuition. Set theory is a foundationalist programme (begun by Cantor and developed by Russell, Zermelo and others); and its aim is to provide an explicit and precise method for characterising acceptable mathematical objects (especially those which are infinite) and acceptable mathematical inference, in terms of axioms about sets or collections alone. Interestingly, it was found that in order to reproduce the main body of mathematical results (for example, results pertaining to the continuum), certain unintuitive axioms had to be accepted. That is, strongly existential, nonconstructive axioms such as the axiom of infinity (there exists an infinite set), the unrestricted power set axiom (for any set there also exists the set of all its subsets), and the axiom of choice (for any set, x, of sets there exists another set which consists of exactly one element from each of the sets in x) had to be accepted in order for set theory to be powerful enough to produce all the mathematical results perceived as important.