From the continuum to large cardinals

Continuity is one of the most important concepts of mathematics and physics. This has been recognized since the time of Euclid and Archimedes, but it has also been understood that continuity raises awkward questions about infinity. To measure certain geometric quantities, such as the diagonal of the unit square, or the area of a parabolic segment, involves infinite processes such as forming an infinite sum of rational numbers. For a long time, it was hoped that “actual” infinity could be avoided, and that the necessary uses of infinity could be reduced to “potential” form, in which infinity is merely approached, but never reached. This hope vanished in the 19th century, with discoveries of Dedekind and Cantor. Dedekind defined the points on the number line via infinite sets of rational numbers, and Cantor proved that these points form an uncountable infinity—one that is actual and not merely potential. Cantor’s discovery was part of his theory of infinity that we now call set theory, at the heart of which is the discovery that most of the theory of real numbers, real functions, and measure takes place in the world of actual infinity. So, actual infinity underlies much of mathematics and physics as we know it today. In the 20th century, as Cantor’s set theory was systematically developed, we learned that the concept of measure is in fact entangled with questions about the entire universe of infinite sets. Thus the awkward questions raised by Euclid and Archimedes explode into huge questions about actual infinity.