Extremality Assumptions in the Foundations of Mathematics

Abstract

The work by J.H. Conway (1976, ch. 4), Philip Ehrlich (1987, with further references) and others on absolute continua derives much of its philosophical and mathematical interest from its instantiating a more general approach to the foundations of mathematics. This approach focuses on the role of extremality (rnaximality and minimality) assumptions as a part of the basis of mathematical theories. The idea of extremality is potentially much more important than has recently been pointed out in the literature. It is illustrated by the principle of mathematical induction, which can be thought of as an attempt to enforce the requirement of minimality on the models of elementary number theory as well as by Hilbert's Axiom of Completeness (1971), which was calculated to enforce a kind of maximality on the models of axiomatic geometry. (These models were of course intended to be a species of continua.)