Cantor's Continuum Problem
Quinn Maurmann
Charcoal and graphite, 2008.
We understand just enough that we can recognize our own failures, and this drives us to the brink of madness. We look upon the infinite, wide-eyed and perhaps crazed with fear and wonder, for we know perfectly that any attempt to understand the infinite is hubris.
A three-page paper in 1874 revolutionized the mathematical conception of infinity. Georg Cantor had demonstrated that it made sense to speak of different sizes of infinity, and in particular he had shown the continuum of all real numbers had to be larger in this sense than the set of integers. His later work further showed that there are sets larger than the real numbers and sets still larger than that and so on; different levels of the infinite all the way up "to infinity."
Pictured is a visual representation of one of Cantor's arguments. The number of nested circles is the first type of infinite, the same type as the integers. However, the set of infinite-length paths desceding ever deeper into these circles is much larger, and turns out to have the same size as the real numbers.
And yet this great triumph led so quickly to defeat. Cantor asked whether there was any set with size between that of the real numbers and the integers, or if the real numbers really were the "next step up." The problem consumed him.
It is now known that Cantor couldn't have ever hoped to solve his problem, for the answer is independent of the established laws of mathematical logic. If mathematics reflects some kind of objective reality, then the problem must have a solution, whether we humans and our finite brains have access to it or not. The alternative seems to be that there is a subjective nature to mathematical truth.
Wherein does the truth lie? Are the patterns on Cantor's eye reflections of images in some objective reality, or are they fantasies projected onto the pupil by the mind?
(For the record, I'll state that I am still a "realist" and believe the problem has a solution, whether human thinkers can find it or not. It could be the most important/deepest problem in all of set theory, and the mathematicians still working on it think it can be resolved by discovering new and overlooked laws of logic.)
Q
