The Axiom of Infinity (2018)
We’ve done an enormous amount of work to make mainstream mathematics work the way it currently works. We have sets which are infinite, but individual elements are not. In other words, you have an infinite set of numbers, but no number is itself infinite. Only after centuries of effort did we make this work rigorously.<p>I think it’s worth remembering this, when we think about alternative foundations for mathematics. What if we discard the axiom of infinity? What if we discard the axiom of choice? Axiom of regularity? Each of these decisions leads us to a path which may be as fruitful as current mainstream mathematics, but which relatively unexplored. Or you could stick with existing formulations, and rework how you do calculus.<p>To pick on the calculus example—mainstream calculus defines continuity using limits, which are defined by formulas quantified over infinite sets. What happens if, instead of using infinite sets, you use sets that contain infinite numbers? Voilà, you have nonstandard analysis… the “dx” and “dy” can now be actual infinitesimal numbers. Is this easier for students to understand? Or is it confusing? Does it formalize existing intuition about infinitesimals, and let you do things like treat dy&#x2F;dx as an actual division?<p>Same is true in the opposite direction. What if you have no infinite sets, no infinite numbers? How do you do calculus? What theorems still work, which don’t?<p>The theorems that don’t work—do we miss them when they are gone?