Intuitive but Non-Rigorous Explanations of Infinite Numbers

Abstract

The mathematical study of infinity seems to have the ability to transport the mind to lofty and unusual realms. Decades ago, I was transported in this way by Rudy Rucker's book Infinity and the Mind. Despite much subsequent learning and teaching of mathematics in the service of physics and astronomy, there remained quite a few aspects of the "higher infinities" that I was still far from comprehending. Thus, I wanted to dive back in to understand those ideas and to find good ways to explain them to other non-specialists. These notes are an attempt to do this. They begin by discussing huge (but still describable) finite numbers, then they proceed to the concepts of fixed points, Cantor's countably infinite ordinals, transfinite cardinals and the continuum hypothesis, and then to more recent attempts to define still-larger infinities called large cardinals. I should warn the reader that the author is an astrophysicist, so these pedagogic explanations may not be satisfying (or anywhere nearly sufficiently rigorous) to a mathematician. Still, I hope these explanations provide some intuitive insights about concepts that are too large to fit in the physical universe.

Figures (10)

• Figure 1: (a) Example iterations $y_n$ from three different initial values of $x$. (b) Multi-valued fixed-point solution $y(x)$, with the stable solutions shown as solid black line, unstable solutions as dotted lines, and the three examples from panel (a) shown with colored points.
• Figure 2: Three infinite ordinals illustrated as telephone poles receding toward the horizon.
• Figure 3: Adding one guest to Hilbert's Hotel.
• Figure 4: Adding $\omega$ guests to Hilbert's Hotel.
• Figure 5: One way to line up the rational numbers in a countable order.